COMPLEX ANALYSIS

MODERN REAL AND COMPLEX ANALYSIS

Bernard R. Gelbaum

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York • Chichester • Brisbane • Toronto • Singapore

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Copyright @ 1995 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012.

Library of Congress Cataloging-in-Publication Data: Gelbaum, Bernard R.

Modern real and complex analysis / Bernard R. Gelbaum. p. em.

"A Wiley-Interscience publication." Includes bibliographical references and index. ISBN 0-471-10715-8 (acid-free) 1. Mathematical analysis. 1. Title.

QA300.G42 1995 515-dc20 94-23715

Printed in the United States of America

10 9 8 7 6 5 4 3 2

Preface

Real analysis and complex analysis are fundamental in modern mathematical education at the graduate and advanced undergraduate levels. The material covered in courses on those subjects has varied over the years. The tendency has been toward periodic revisions reflecting a changing consensus in the mathematical community. The aim of this book is to present a modern approach to the subjects as they are currently viewed.

The reader is assumed to be familiar with such terms as continuity , pow er s eri es , uniform conv erg en ce and uniform continuity , d erivativ e, Ri emann int egral , etc. On the other hand, a considerable effort has been made to provide, in the union of the text proper, the SYMBOL LIST, and the GLOSSARY/INDEX, complete definitions of all mathematical concepts introduced. The following notations obtain for assertions in formal logic:

{A} :::;.. {B} for A implies B {A} <* {B} for A iff B A /\ B : for A and B

A V B for A or B.

Among the novel and unique features of the text are the following.

In Chapter 1.

a) Topology discussed three ways: via open sets, via nets, and via filters.

b) Two proofs of Brouwer's Fixed Point Theorem. c) Uniform spaces.

In Chapter 2.

a) Integration viewed as a Daniell functional. b) A detailed exploration of the connection between measure as de

rived from a Daniell functional and classical Lebesgue-Caratheodory measure.

c) The Riesz Representation Theorem as a consequence of Daniell's approach.

v

VI Preface

In Chapter 3.

a) Functional analysis and weak topologies. b) Banach algebras. c) Axiomatics of Hilbert space and linear operators.

In Chapter 4.

a) The Fubini-Tonelli Theorems via Daniell's techniques. b) A unified approach to nonmeasurable sets. c) Differentiation by direct methods that avoid parameters of regu

larity, nicely shrinking sequences, etc. d) Haar measure by Daniell functionals.

In Chapter 5.

a) Singular homology of the plane via the formulre and theorems of Cauchy.

b) Elementary exterior calculus as applied to complex function theory.

In Chapter 6.

a) Subharmonic functions, barriers, and Perron's approach to Dirichlet's problem.

b) Poisson's kernels and approximate identities in Banach algebras. In Chapter 7. a) Runge's Theorem and its application to Mittag-Leffler's Theorem;

the latter as a source of WeierestraB's product representation. b) Entire functions and their orders of growth.

In Chapter 8.

a) Riemann's Mapping Theorem and its connection to Dirichlet's problem.

b) Bergman's kernel functions and conformal mapping. c) Automorphic functions. d) Green's functions.

In Chapter 9.

a) Picard/Montel Theorems and their consequences. In Chapter 10.

a) A thorough treatment of analytic continuation. b) The Riemann-WeierstraB-Weyl concepts of Riemann surfaces as

well as Riemann surfaces defined as connected one-dimensional complex analytic manifolds.

c) Covering spaces, sheaves, lifts. d) The General Uniformization Theorem derived via a sequence of

carefully graded Exercises. In Chapter II. a) Thorin's Theorem. b) Applications to M. Riesz's Convexity Theorem and related parts

of functional analysis. In Chapter 12. An introduction to the theory of complex functions of more than one complex variable.

Preface vii

Within each section, all the numbered items save Figures, e.g. , TfIEOREMS, Exercises, equations, are numbered consecutively as they appear. Thus in Section 3.2 , the first item is 3.2.1 LEMMA, the second item is, 3.2.2 COROLLARY, the third item is (3.2.3), (the first) numbered equation, etc. State University of New York at Buffalo B. R. G.

Contents

REAL ANALYSIS

1 Fundamentals 3

1.1 Introduction 3

1.2 Topology and Continuity 6

1.3 Baire Category Arguments 18

1.4 Homotopy, Simplices, Fixed Points 20

1.5 Appendix 1: Filters 32

1.6 Appendix 2: Uniformity 34

1.7 Miscellaneous Exercises 38

2 Integration 44

2.1 Daniell-Lebesgue-Stone Integration 44

2.2 Measurability and Measure 55

2.3 The Riesz Representation Theorem 76

2.4 Complex-valued Functions 79


3 Functional Analysis 89

3.1 Introduction 89

3.2 The Spaces LP, 1 ::; p ::; 00 92

3.3 Basic Banachology 103

3.4 Weak Topologies 111

3.5 Banach Algebras 116

3.6 Hilbert Space 128


IX

x Contents

4 More Measure Theory 137

4.1 Complex Measures 137

4.2 Comparison of Measures 142

4.3 LRN and Functional Analysis 147

4.4 Prod uct Measures 151

4.5 Nonmeasurable Sets 158

4.6 Differentiation 164

4.7 Derivatives 177

4.8 Curves 185

4.9 Appendix: Haar Measure 187


COMPLEX ANALYSIS

5 Locally Holomorphic Functions 203

5.1 Introduction 203

5.2 Power Series 205

5.3 Basic Holomorphy 209

5.4 Singularities 230

5 .5 Homotopy, Homology, and Holomorphy 245

5.6 The Riemann Sphere 251

5.7 Contour Integration 254

5.8 Exterior Calculus 257


6 Harmonic Functions 270

6.1 Basic Properties 270

6.2 Functions Harmonic in a Disc 273

6.3 Subharmonic Functions and Dirichlet's Prohlem 284

6.4 Appendix: Approximate Identities 296


7 Meromorphic and Entire Functions 701

7.1 Approximations and Representations 301

7.2 Infinite Products 313

7.3 Entire Functions 323


Contents

8

9

10

11

12

Conformal Mapping

8.1 Riemann's Mapping Theorem 8.2 Mobius Transformations 8.3 Bergman's Kernel Functions 8.4 Groups and Holomorphy 8.5 Conformal Mapping and Green's Function 8.6 Miscellaneous Exercises

Defective Functions

9.1 Introduction 9.2 Bloch's Theorem 9.3 The Little Picard Theorem 9.4 The Great Picard Theorem 9.5 Miscellaneous Exercises

Riemann Surfaces

10.1 Analytic Continuation 10.2 Manifolds and Riemann Surfaces 10.3 Covering Spaces and Lifts 10.4 Riemann Surfaces and Analysis 10.5 The Uniformization Theorem 10.6 Miscellaneous Exercises

Convexity and Complex Analysis

11.1 Thorin's Theorem 11.2 Applications of Thorin's Theorem

Several Complex Variables

12.1 Survey

Bibliography

Symbol List

Glossary /Index

xi

336

336

342

349

357

363

365

369

369

371

374

376

380

381

381

391

409

419

422

424

431

431

434

445

445

451

455

461

MODERN REAL AND COMPLEX ANALYSIS

REAL ANALYSIS

1 Fundamentals

1.1. Introduction The text is addressed to readers with a standard background in under

graduate algebra, analysis, and elementary topology. Hence, when reference is made to concepts such as groups, maps, posets, topological spaces, etc. , there is an underlying assumption that the reader is familiar with them. Nevertheless, all terms and notations essential for the understanding of the material are defined or explained in the Chapters, the GLOSSARY /INDEX, or the SYMBOL LIST.

Real analysis deals with the study of functions defined on a set X and taking values, for some n in N, in the set ffi.n of n-tuples of real numbers or occasionally in the set Cn of n-tuples of complex numbers.

On the other hand, complex analysis is confined to the study of locally holomorphic functions, i .e . , for some nonempty connected open subset, i.e. , region, Q of C, functions f in Cn and differentiable throughout Q.

Beginning with a system, e.g. , that of Zermelo-Fraenkel, (ZF) [Me] , of axioms for set theory, one can construct in turn the system N �f {I, 2, . . . } of natural numbers , the ring Z of integers , the field Q of rational numb ers, and finally the field ffi. of real numbers [La] . The result is a field endowed with an order < (a transitive relation such that for any two (different) elements x and y, precisely one of x < y and y < x is true) ; ffi. is complete with respect to <, i .e . ,for every subset bounded above there is a unique supremum (least upper bound) . Since any two complete ordered fields are field- and order-isomorphic [0] , one may proceed directly as follows.

1 .1 .1 DEFINITION. THE SET ffi. IS A COMPLETE ORDERED FIELD. The multiplicative identity 1 of ffi. gives rise to 1 , 1 + 1 , . . . , i.e., to the

set N of natural numbers. The ring Z of integers is the set of R-equivalence classes of ffl :

{ (m, n)R (m' , n') ) {} {m + n' = m' + n} .

The field Q of rational numbers is the set of all S-equivalence classes of Z x (Z \ {O} ) :

{ (p, q) S (p', q' ) } {} {pq' = p' q } .

3

4 Chapter 1. Fundamentals

The field C of complex numbers is the set

JR.2 �f { (a, b) : {a, b} c JR. } �f JR. x JR.

in which the algebraic operations addition ( +) and multiplication ( . ) as well as the symbols 0 and i are defined according to:

( a, b) + (c, d) �f (a + c, b + d) , (a, b) . (c, d) � (a, b) (c, d) �f (ac - bd, be + ad) ,

o �f (0, 0) , i �f (0, 1 ). Furthermore, JR. is identified with JR. x {O} and then (a, b) � a + ib. When z �f a + ib � ?R(z) + i�(z) E C, the absolute value of z is I z l �f J a2 + b2 . (When a E JR. and a is regarded as a + iO, an element of C, the definition of l a l as just given and the definition

l a l �f {a -a if a 2': 0 if a < 0

clef . clef a - ib are equivalent . ) If z = a + zb -j. 0 and w = W' wz = zw = l .

[ 1 .1 .2 Note. Below and throughout the book, t o avoid imperatives, most Exercises are phrased as assertions to be proved.]

1 . 1 .3 Exercise. a) The set N is the intersection of all JR.-subsets S containing 1 and such that if X E S, x + 1 E S. b) The order in JR. is Archimedean, i.e. , if E > 0 and M > 0, then for some n in N, nE > M.

[Hint: b) Otherwise, for some positive E and M and each n in N, nE :s; M, whence { nE : n E N } has a supremum.]

1.1.4 Exercise. If a, b E JR., then la + b l :s; l a l + I b l. Equality obtains iff for some nonnegative constants A, B, not both zero, Aa = Bb . 1 .1 .5 Exercise. a) Addition and multiplication in C are commutative operations. b) If a + ib E C, l a + ib l :s; l a l + Ib l and equality obtains iff ab = o.

The following special types of subsets of JR. and C appear frequently in the text :

• when -00 :s; a :s; b :s; 00, the oriented real intervals: (a , b) �f { X a < x < b } , (open) ,

[a , b) �f { x a:S;x < b } , (right-open) ,

(a, b] �f { x a < x:S;b } , (left-open) ,

[a, b] � { x a:S;x:S;b } , (closed) ;

Section 1.1. Introduction

• when S c JR., S+ � sn [0, (0) ; • when {p , q} c e the oriented compl ex intervals :

(p, q) �f { z [p, q) �f { z (p, q] �f { z [p, q] �f { z

z = (1 - t)p + tq, 0 < t < I } , (open) ,

z = (1 - t)p + tq, 0 ::; t < I } , (right-open),

z = (1 - t)p + tq, 0 < t ::; I } , (left-open) ,

z = (1 - t)tp + tq, 0 ::; t ::; 1 } , (closed) ;

( [a, b) , (a, b] , [p, q) , and (p, q] are half-open complex intervals) ;

5

• the subgroup 11' �f { z : z E e, Iz l = I } of the multiplicative group of nonzero elements of C. The (possibly empty) interior of any of the real intervals above is ( a, b) . For a set {Xl' } 'YEr' X 'YEr Xl' is the Cartesian product of the sets Xl"

i.e. , X'YErX'Y �f { f : f : r 3 "I f--t f b) E X'Y } . Since fh) E Xl" occasionally the notation xl' is used for f h) and an f is a vector {xl' } 'YEr When, for some X, Xl' == X, then X'YEr Xl' = Xr, the set of all maps from r into X.

An n-dimensional interval I in JR.n is either the empty set (0) or the Cartesian product of n one-dimensional intervals each of which has a nonempty interior. For n in N, a half-open n-dimensional interval in

is the Cartesian product X:= l [ak , bk ) of right-open intervals. If bk - ak is k-free, the half-open n-dimensional interval is a half -open n-dimensional cube . When n > 1, elements of JR.n or en are regarded as vectors and are denoted by boldface letters : a, x, . . . . The vector (0, . . . , 0) is denoted o.

clef ( ) . I II clef The length or norm of the vector x = Xl , ... , Xn IS I x 2 = n

k=l The cardinality of X is denoted # (X) , e.g. , # (N) �f No, # (JR.) �f c.

The ordina l number of the well-ordered set of equivalence classes of wellordered countable sets is f2. (The previously introduced use of Q-to denote a region-causes no difficulty since the two contexts-ordinal numbers and regions-do not occur together in the remainder of the book. )

When n is a cardinal number the phrase-n objects-means there is a set S consisting of pairwise different objects and # ( S) = n. Thus, the phrase-two points x and y-implies x i- y.

On the other hand, the phrase-the points x and y-carries no such implication: both x = y and x i- y are admissible.


1.2. Topology and Continuity

1.2.1 DEFINI TION. A TOPOLOGICAL SPACE IS A SE T X PAIRED WI TH A SUBSE T T OF !fj(X) , THE SE T OF ALL SUBSE TS OF X. THE SE T T IS THE TOPOLOGY OF X AND THE ELEMEN TS OF T ARE THE OPEN SE TS OF X. THE AXIOMS GOVERNING T ARE:

a) (/) E T AND X E T; b) T I S CLOSED WI TH RESPEC T TO THE FORMA TION OF ARBI TRARY

UNIONS AND FINI TE IN TERSEC TIONS .

The set of open sets of X is also denoted O(X) . When A C X, a ) and b ) hold for the set

clef { } TA = A n U : U E T ,

which endows A with the relative topology induced by T.

1.2.2 Exercise. For a topological space X, the set F(X) of complements of elements of O(X) is governed by: a') (/) E F(X) and X E F(X) ; b') F(X) is closed with respect to the formation of arbitrary intersections and finite unions.

When T and T' are topologies for X and T C T', T' is stronger than T while T is weaker than T'.

1 .2.3 Example. For any set X there are: a) the strongest or discrete topology !fj(X) consisting of all subsets of X ; b) the weakest or trivial topology consisting only of (/) and X .

1 .2.4 Example. For JR., the customary topology consists of all (arbitrary) unions of open intervals . Unless the contrary is stated, JR. is regarded as endowed with its customary topology.

On the other hand, the Sorgenfrey topology T s for JR. consists of all (arbitrary) unions of left-closed intervals, i .e . , unions of all sets of the form

[a, b) �f { x : JR. 3 a ::; x < b E JR. } .

For a topological space X a subset B �f {U), LEA of T is a base for T iff every element (open set) in T is the union of (some) elements of B. 1 .2 .5 Example. In both the usual and Sorgenfrey topologies for JR., Q meets every open set.

The countable set { (a, b) : Q 3 a < b E Q} is a base for the customary topology. By contrast, if B is a base for Ts and [a, b) E Ts , a must

Section 1.2. Topology and Continuity 7

belong to some base element B contained in [a, b) . Thus # (8) 2: # (JR.) : there is no countable base for T s.

If 8 is an arbitrary subset of !fj(X), 8 is contained in the discrete topology !fj(X): the set of all topologies containing 8 is nonempty. The intersection T B of all topologies containing 8 is the topology for which 8 is a base: 8 generates T B.

When, for (X, T), there is a countable base for T, X is second countable . When X contains a countable subset meeting every element of O(X), X is separable . 1 .2.6 Example. The set { (a, b) n : {a, b} c Q, a < b } is a countable base for the customary topology for JR.n .

When X and Y are sets and I E yX , I is:

• injective iff {J(a) = I(b) } :::;. {a = b} ;

• surjective iff I(X ) = Y; • bijective iff I is injective and surjective ; • autojective iff X = Y and I is bijective.

Injective, surjective, . . . maps are injections, surjections, . . . When (Xl, Td and (X2, T2) are topological spaces and I E X:;l (the

set of all maps from Xl into X2), I is:

• continuous iff 1-1 (T 2) c T 1; • open iff I (T d c T 2· • a homeomorphism iff I is bijective and I is both continuous and open,

i .e . , iff I is bijective and both I and 1-1 are continuous.

The set of continuous maps in X:; 1 is denoted C (Xl, X 2) . 1.2.7 Exercise. If (I, g) E C(X, Y) x C(Y, Z) , then g o I E C(X , Z) .

When A C X and O(A) is the set of open subsets of A, AO �f U U,

UEO(A) is the (possibly empty) interior of A. For a nonempty subset A of X, a neighborhood N(A) is a set such that A C N(At . For simplicity of no -tation, when x E X , N (x) �f N ( { x } ) . The set of neighborhoods of A is N(A).

1 .2.8 Exercise. If A i- 0, for F �f N(A) , the following obtain: a) F i- 0, ° tJ. F; b) {F, F' E F} :::;. {F n F' E F}; c) {{F E F} 1\ {F c Gn :::;. {G E F}.


A base of neighborhoods at a point x in a topological space X is a set B such that: a) B e N(x) ; b) every N(x) is a union of elements of B. A base of n eighborhoods for X is a set B such that : c) each element is a neighborhood of some x in X; d) each neighborhood N(x) of each x in X is a union of (some) elements of B. 1 .2.9 Example. A metric space is a set X paired with a map

d : X2 3 (x, y) f--t d(x, y) E [0, (0)

such that a) d(x, y) = 0 iff x = y and b) d(x, z) ::; d(x, y) + d(z, y) (whence d(x, y) = d(y, x ) ) . The notation (X, d) is analogous to (X, T ) . When x E X and r � 0, the open (closed) ball B(x, rt (B(x, r ) ) centered at x and of radius r is { y : d(x, y) < r } ({ y : d(x, y) ::; r }) . The metric d induces a topology T for which a base is the set of all open balls and a base of neighborhoods at x is the set of all open balls centered at x. Furthermore, (X, T) is a Hausdorff space and T is a Hausdorff topology, i.e. , if x and y are two elements of X, there is a neighborhood N(x) and a neighborhood N(y) such that N(x) n N(y) = 0.

1.2.10 Example. The function

is the customary(Euclidean} metric for JR.n . The sequence S � {xn}nEN contained in (X, d) converges iff for some x in X, lim d (xn , x) = 0; when n--+=

lim d (xm , xn ) = 0 , S is a Cauchy sequence . The metric space is com -m,n----tCXJ plete iff each Cauchy sequence {xn}nEN converg es .

The next discussion i s facilitated by the introduction of the extended - clef real number system: JR. = JR. U {-oo} U {oo} and the use of extended JR. -

valued functions, i .e., functions that may assume the "values" ±oo. The topology of ffi: is determined by the neighborhood base consisting of all neighborhoods in JR. together with the following sets of subsets of JR.:

• the complements of all closed sets bounded above and unbounded below (the neighborhoods of 00 )

• the complements of all closed sets bounded below and unbounded above (the neighborhoods of -(0).

In ffi:, limiting operations admit the possibility that limits can be ±oo. Thus, e.g. , if {xn} EN, and Xn ::; Xn+ l , n E N, lim Xn is either a number n n n---+CXJ (in JR.) or 00 . In the latter instance, for each N (00 ) , there is an no such that if n > no, Xn E N (00) . Similar observations apply when other relevant operations in ffi: are encountered.

Section 1.2. Topology and Continuity

-x For f in ffi. , a in X, and N(a) ,

lim f(x) �f inf sup f(x) , lim f(x) �f sup inf f(x) . x=a NEN(a) xEN x=a NEN(a) xEN

9

In JR:, 0 . (±oo) �f o. When {an} nEN is a monotone sequence contained in ffi., lim an exists and is in JR:. n-+=

[ 1 .2 .11 Note. Neither of the possibilities

lim f(x) = ±oo, lim f(x) = ±oo x.=a is excluded.]

According as lim f(x) = f(a) resp. lim f(x) = f(a) , f is upper semi-x=a x=a continuous (usc) resp. lower semicontinuous (lsc) at a; when f is usc resp. lsc at each a in some set S, f E usc(S) resp. f E Isc(S) .

1.2 .12 Exercise. A function f in JR:x i s usc resp. lsc at a iff for each oc in JR:, { x : f(x) < oc} resp. { x : f(x) > oc} is open.

[Hint: If f E usc(X) and f(x) < oc, there is an N (x) such that sup f(y) < oc. If f tJ. usc(X) , for some x and some positive YEN(x)

E, f(x) < t� f(y) - 2E. Then { z : f(z) < ti:! f(y) - E } is not open, since otherwise there emerges the contradiction:

limy = x)f(y) ::; lim f(y) - E < lim f(y) .] ( y=x y=x

When l1 is a subset of r and for each r5 in l1, A8 is a subset of X8 ,

3 ({A8 }8E�) �f X8E�A8 X X'YEr\�X'Y

is a cylinder. When # (l1) E N, 3 ({A8}8E�) is a finite cylinder. When each Xl' is a topological space, an element B of a base for the

product topology of X 'YEr Xl' is determined by:

a finite subset bl' . . . ' 'Yn } of r ,

points X'Yi in X'Yi' 1 ::; i ::; n, neighborhoods N 1'1 (X'Y1 ) , ... , N 'Yn (x'Yn ) ,

and B�f Bbl, . . . ,'Yn;x'Y" ... ,x'Yn;N'Y1 (x'YJ, ... ,N'Yn (x'YJ ] is the finite cylinder determined by the finite subset l1 �f {'Yl , . . . , 'Yn} of r and neighborhoods N 'Yi (x'YJ , 1 ::; i ::; n. The set of all finite unions of such cylinders is closed with respect to the formation of finite intersections.


1.2.13 Example. For n in N, the customary topology for ffi.n is the product topology derived from the customary topology for ffi..

1 .2.14 Exercise. For n E N, p fixed in (0, (0) , x �f (Xl,"" Xn) a typical element of ffi.n, and

BP(x, r)O �_ef { y � ( )P clef II l iP } : � Xk - Yk = X - Y P < r , k==l

the set { BP (X, r ) ° : x E ffi.n, 0 < r } is a base for the product topology of ffi.n .

1.2 .15 Exercise. For a set {X'Y}'YEr of topological spaces, if A E r, the . t' X clef X X clef { } X . . proJec zan 1l'A : = 'YEr I' 3 x = xl' H XA E A Is a contmuous open

map with respect to the product topology for X. A partially ordered set (poset) i s a pair (r �f b}, -<) (or simply r) in

which the order -< is transitive and the relation "/ -< ,,/' (also written ,,/' >- ,,/) obtains for a (possibly empty) set of pairs b, ,,/') . The partially ordered set r is directed iff for each pair (,,/, ,,/') in r2 , for some "/" in r, "/" >- "/ and "/" >- ,,/'. A directed set r is a diset and a subdiset of r is a subset that is a diset with respect to the partial order in r 1.2 .16 Example. For any set 5 the set <1>(5) consisting of the empty set (/) and all finite subsets of 5 is a diset with respect to inclusion as a partial order -<: A c B c 5 {} A -< B.

1.2 .17 Example. For a point X in a topological space X, the set N(x) of all neighborhoods N (x) of X is a diset with respect to the partial order of reversed inclusion: Nl (x) -< N2 (x) {} N2 (x) C Nl (x) .

1.2 .18 DEFINI TION. A MAP n : A 3 A H n(A) E X OF A DISE T

A � {A, -<}

IS A NE T. WHEN X IS A TOPOLOGICAL SPACE, THE NE T n CONVERGES TO a IN X IFF FOR EACH N(a) AND SOME Ao [N(a)] IN A,

{Ao [N(a)] -< A} :::;. {n(A) E N(a) } .

WHEN X = ffi. ,

n � limn(A) �f inf sup n(A) and limn(A) � sup inf n(A) . AEA /LEA A';-/L AEA /LEA A';-/L

A point a is a limit point of a set A (a E A· ) iff every N (a) meets A \ {a} . A point a in A is isolated iff for some N ( a ) , N ( a) n A = {a}.

Section 1.2. Topology and Conti nuity 11

The closure A of A is A U A-and A is closed iff A = A. A subset D

of a topological space X is dense (in X) iff D = X. The boundary 8(A) of A is An X\A.

1.2 .19 Exercise. A subset A of a topological space X is: a) closed iff the complement X \ A is open; b) dense iff A meets every member of O(X). Furthermore,

c1) A=A; c2) Au B = A U B; c3) A C A; c4) 0 = 0; c5) A- = A-; c6) (A-r c A-.

Subsets A and B of X are separated iff (A n B) U (A n B) = 0. The space X is connected iff X is not the union of a pair of separated and nonempty sets, i .e . , iff X contains no nonempty proper subset that is both open and closed, i.e. , iff X is not the union of two nonempty and disjoint open subsets, i.e. , iff X is not the union of two nonempty and disjoint closed sets.

When C is regarded as ffi.2 with the product topology, a nonempty connected open subset of C is a region, usually denoted fl, the same symbol introduced earlier for a special ordinal number.

1.2.20 Exercise. The only connected subsets of ffi. are intervals, i .e., -<

denoting < or ::;, sets of the form { x : -00 -< a -< x -- Ao} :::;. {n(A) E A} ;

n is frequently in A iff for every A, there is an a N such that A' >- A and n (A') EA. Two nets n : A 3 A f--t X and n' : A' 3 A' f--t X are essentially equal (n � n') iff for some AO resp. A� in A resp. A',

{ { A >- AO} !\ {N >- A�}} :::;. {n ( A) = n (A' ) } ;

� is an equivalence relation. When A is a diset and A' C A, A' is cofinal with A iff for each A in

A, there is in A' a N (A) such that A' (A) >- A. If A' is cofinal with A and n : A -+ X is a net , the net n!A" the restriction of n to A', is cofinal with n.

1.2.21 Exercise. a) a E A iff some net n with range in A converges to a; b) a E A -iff some net n with range in A \ {a} converges to a; c) A is closed iff A- C A; d) a is an isolated point of A iff a E A \ A-.


1 .2.22 Exercise. If, for some x, the net n : A H X, is frequently in each N(x), A contains a subdiset r such that some net m : r H X converges to x.

1 .2.23 DEFINI TION . A TOPOLOGICAL SPACE X IS compact IFF FOR EVERY SE T {Un}nEA OF OPEN SE TS S UCH THA T U Un = X, THERE IS A FINI TE

nEA n

SUBSE T {Unk h<k<n SUCH THA T U Unk = X. (Every open cover admits k=l

a finite subcover. ) A SUBSE T K OF X IS COMPAC T IFF K IN I TS induced topology IS A COMPAC T SPACE. THE SPACE X IS locally compact IFF SOME NEIGHBORHOOD BASE FOR X CONSIS TS OF COMPAC T SE TS.

The set of compact subsets of a topological space X i s denoted K(X) . [ 1.2.24 Remark. Analogously, X i s locally connected iff some neighborhood base for X consists of connected sets. In Chapter 10 locally curve -connected spaces are defined. They play a role in the study of Riemann surfaces , v. Chapter 10.]

1 .2.25 Exercise. If:F �f {F>,} ),EA is a set of closed subsets of a compact space X and the intersection of any finite subset of :F is nonempty, n F), -j. 0. (The set :F enjoys the finite intersectio n property (fip) . ) ),EA 1 .2.26 Exercise. a) A closed subset of a compact set is compact . b) A compact subset of a Hausdorff space is closed.

1.2.27 Exercise. If X is compact and f E C(X, Y) , f(X) is compact.

1 .2.28 Exercise. If X is a locally compact Hausdorff space and

K(X) 3 K C U E O(X) ,

for some V in O(X) , V i s compact and K e V e V c U. In particular, X is locally compact iff each N (x) contains a compact neighborhood of x.

[Hint: Each x in K is in some compact neighborhood N (x) contained in U.]

1 .2.29 LEMMA. A HAUSDORFF SPACE X IS LOCA LLY COMPAC T IFF FOR EACH x THERE IS SOME N (x) SUCH THA T N (x) IS COMPAC T.

PROOF. If X is locally compact the conclusion above follows by definition. Conversely, if N1 (x) is given, there is some N2 (x) such that N2 (x) is

compact , whence N (x) � N1 (x) n N2 (x) is a compact neighborhood of x. However, owing to 1 .2 .19, N (x) = N (x). 0


1 .2.30 THEOREM. (Tychonov) IF {X>.} >'EA IS A SE T OF COMPAC T SPACES, X �f II X>. IS COMPAC T (WI TH RESPEC T TO I TS PRODUC T TOPO LOGY ) .

>'EA

PROOF. Among all fip subsets 9 �f {Gs LEs of!fj(X) there is a partial order -< defined according to the rule: {91 -< g2} {} {91 C g2}. If:F �f {Fp} pEP is a set of closed subsets of X and :F enjoys the fip, Zorn's Lemma implies that :F is contained in a set gmax that is maximal with respect to the given partial order. The following obtain: a) if A E gmax and B E gmax, then An B E gmax; b) if A c X and A meets each element Gs of gmax, A E gmax'

The projection 11"1' : X 3 ( ... , x>., . . . ) f--t xl' E XI' engenders the set

of closed subsets of XI'" Each gmax,JL enjoys the fip. Hence, if , for each . II- ( ) clef fJ m A, xl' E 11"JLGs , then ... , xI"'" = x E X. If N(x) is a (prod-sES

uct topology) neighborhood of x, for some finite subset P'I,"" An} of A n

and neighborhoods N>'i (x>.J , 1:::; i :::; n, N(x) = n 11").,1 [N>'i (x>.J]. Since, i=1

for each s in 5, N>'i (x>.J meets 11">'i (Gs ) , it follows that for each s in 5, 11").i1 [N>'i (x>.J] meets Gs , whence 11").i1 [N>'i (x>.J] E gmax' Hence N(x) meets each Fp: for each p, x E Fp = Fp, i.e., n Fp i- 0. 0

1 .2.31 Exercise. For the Cantor set

pEP

{ =

} clef En Co = L 3n : En = ° or En = 2 n=1

= = and the map ¢ : Co 3 L �: f--t L 2�:1' it follows that ¢ (Co ) = [0,1].

n=1 n=1

1 .2.32 Exercise. If X and Yare topological spaces and T E Y x, T is open iff for each x in X and each neighborhood N(x) there is a neighborhood N[T(x)] contained in T[N(x)].

1.2.33 Exercise. If ¢ is a monotonely inc reasing function in ffi.1R,

# [Discont (¢)] :::; No .

[Hint : To each discontinuity a of ¢ there corresponds the nonemp

ty open interval (lim ¢( x), lim ¢( x)) , which meets Q.] xta x.j.a


1.2.34 Exercise. In C(JR., JR.) there is no function f such that for each a in JR., # [J-l(a)] = 2.

1.2.35 Exercise. If f E C(JR., JR.) and f(x + y) == f(x) + f(y) , for some A, f(x) == Ax.

[Hint: If n E N, f(nx) = nf(x) ; A = f(l ) .]

1.2.36 Exercise. a) If (X, d) is a metric space, F E F(X) , K E K(X), and F n K = 0, J �f inf d(x, y) > 0. b) If

xEF,yEK

Fl � { (x, y)

F2 �f { (x, y)

(x, y) E JR.2 , xy = I } ,

( x, y) E JR.2 , Y = 0 } , then Fl and H are disjoint closed subsets of JR.2 and inf d(x, y) = o.

xEFl,yEF2

[Hint: a) If

{xn}nEN C K, S �f {Yn}nEN C 8(U), and d (xn , Yn) + 0, then S contains a convergent subsequence.]

1.2.37 Exercise. If F �f {Fi}�=l is a finite set of pairwise disjoint closed subsets of a compact metric space (X, d) , there is a positive constant c(F) such that for each x in X there is a set Fi (x) for which

d (x, Fi(x) ) �f inf d(x, y) � c(F) . yEF,(x)

[Hint: For n in N, the (possibly empty) sets

clef { 1 } Gn = x : sup d (x, Fi) ::; - , n E N l �i�n n

are closed and Gn ::) Gn+1• If the conclusion is false, some x is in n

n Gn and x E n Fd nEN i=l

1 .2.38 Exercise. For a finite set F �f {Fi } �= 1 of closed: ..!bsets of a compact metric space (X, d) , there is a positive J(F) such that if

diam (S) � sup { d(x, y) : {x, y} C S } < J(F)

k and S meets each of Fi" . . . , Fik , then n Fik -j. 0.

i=l


[Hint: For each subset Fi of pairwise disjoint elements of F, c (Fi ) > ° and 8(F) = inf c (Fi ) .]

t

1.2 .39 Exercise. (Lebesgue's covering lemma) For an open cover

of compact metric space (X, d) : a) there is a positive l1(U) such that each x in X is in some Ui while d [x, (X \ Ui) ] 2': l1(U) ; b) if S e X and diam (S) < l1(U) , S is a subset of some Ui .

[Hint: If Fi �f X \ Ui and F �f {FiL�i�n ' l1(U) = 8(F).]

[ 1 .2 .40 Note. The numbers 8(F) resp. l1(U) are the Lebesgue numbers of F resp. U.]

For a topological space X and an f in eX, the support of f is

supp (f) �f { x : f(x) -=j:. O } .

When K is a compact subset of an open set V in a topological space X, f E C(X, ffi.) , and f(X) C [0, 1] , the notation K -< f resp. f -< V signifies f(K) = { l } resp. supp (f) C V. 1.2.41 LEMMA. (Urysohn) IF K IS A COMPAC T SUBSE T OF AN OPEN SUBSE T V OF A LOCAL LY COMPAC T HAUSDORFF SPACE X, FOR SOME CON TINUO US f, K -< f -< V. PROOF. Since X is locally compact, K may be covered by finitely many compact neighborhoods contained in V, whence K is contained in an open set U such that U is compact, i.e. , V may be assumed to be compact .

Thus Fa �f 8(V) and Fl �f K are disjoint compact (hence closed) subsets of the compact set V. For x fixed in Fa and each Y in Fl there are disjoint open neighborhoods Ny(x) and Nx(Y). Hence there are neigh-borhoods Nx (Yi) , 1 ::; i ::; n, such that Fl c U Nx (Yi) �f U1• Then

W(x) � n Ny, (x) is open and W(x) n Ul = 0. It follows that l�i�n

Ua �f U W(x) xEFo

is open, Ua n Ul = 0, whence Ua n U1 = Ua n U1 = 0. Similarly, the pairs


of disjoint compact sets can be treated to yield open sets Upq, p, q E {O, I} such that Po c Uoo, (V \ Uo) C UOI, Uo C UlO, PI CUll, Upq n Upfqf = (/) if (p, q) i- (p' , q'). By induction, for each finite sequence (p, q, r, . . . ) of zeros and ones ( binary markers) there is an open set Upqr . . . . Each finite sequence (p, q, r, . . . ) of binary markers defines a dyadic rational number

clef p q r t = - + - + - + · · · 2 22 23

in [0, 1 ] . If t < t', then U(t) �f Upqr ... C Upfqfrf ... C Upfqfrf . . . � U (t' ) . The function

f : X01X H f(x) = { 0 { sup t

is continuous and K -< f -< V.

if X tJ- V x tJ- U(t) } if x E V '

1.2 .42 Exercise. The last two sentences above are valid. o

1 .2 .43 DEFINI TION . FOR AN OPEN COVER U �f {U>.}),EA OF A TOPOLOGICAL SPACE X, THE FUNC TIONS IN <t> � {<P),hEA ARE subordinate TO U IFF <P)' -< U)' , '\ E A. THE SE T <t> IS A partition of unity IFF, TO BOO T, FOR EACH '\: a) <P)' :2: 0, b) FOR EACH x IN X, ONLY FINI TELY MANY <p), (x) i- 0, AND c) L <p), (x) == 1 .

),EA

1.2 .44 THEOREM. IF K IS A COMPAC T S UBSE T OF A LOCALLY COMPAC T clef { } HAUSDORFF SPACE X AND U = U), ),EA IS AN OPEN COVER OF K, FOR K REGARDED AS A TOPOLOGICAL SPACE , THERE IS A PAR TI TION OF UNI TY <t> �f {<P),hEA CON TAINED IN C(X, ffi.) AND SUBORDINA TE TO U.

PROOF. Each x in K is in some U), and belongs to a compact neighborhood N(x) contained in U),. For some finite set {xih�i�n' K C U N (X i) .

h clef For eac '\, K), = U N (Xi ) is a compact subset of U), and 1 .2.41 N(x,JcU),

implies there is a g)' such that K), -< g), -< U),. Since there are only finitely many pairwise different K)" there are only finitely many g)'. At each x . ( ) clef", ( ) d clef {g),} clef { } h III K, S x = � g)' X > ° an <t> = S = <P)' ),EA meets t e

),EA ),EA requirements. 0 1.2 .45 Exercise. The last sentence above is valid.

1.2.46 THEOREM. (Dini) HYPO THESIS : X IS COMPAC T; A IS A DISE T; n IS A MAP n : A x X 3 ('\, x) H n('\, x) E ffi. SUCH THA T:


a) FOR EACH '\, n('\, X) E C(X, JR.) ; b) FOR EACH x, n('\ , x) IS A NET, {A -< IL}:::;" {n('\, x) � n(lL, X) } , AND

n('\, x) CONVERGES TO ZERO .

CONCLUSION: IF E > 0, FOR SOME '\(E) , IF ,\ >- '\(E) AND x E X, THEN n('\, x) < E. PROOF. If the conclusion is false, for some positive E and each ,\ in A, K>. �f { x : n('\, x) � E } is closed and nonempty. The hypothesis implies that {K>.hEA enjoys the fip, whence S �f n K>. -j. 0. For x in S and each

>'EA ,\ in A, n('\, x) � E, a contradiction. 0

[ 1.2.47 Note. Paraphrased, Dini 's Theorem says that a diset of nonnegative continuous functions converging monotonely to zero on a compact space converges uniformly to zero.]

1.2.48 Example. If X is compact, {fn}nEN C C(X, JR.) , and fn + 0,

fn �O.

[ 1.2.49 Note. For a topological space (X, T) , a subset A of X in its induced topology T A, and a subset S of A, topological properties such as openness, closedness, connectedness, etc. , may differ for S according as S is viewed as a subset of A or as a subset of X.

Thus, if X = JR., A = (0, 00) and S = (0, 1 ] , then S is a relatively closed subset of A and is not a closed subset of X. Similarly if A = [0, (0) and S = [0, 1 ) , S is a relatively open subset of A and is not an open subset of X . ]

1.2.50 Exercise. Compactness is an absolute topological property : if (X, T) is a topological space, A C X, and S e A, S is T A-compact iff S is T-compact.

[ 1.2.51 Remark. Owing to 1.2.49, the following locution IS entrenched in the language of mathematics.

The subset S of A (contained in X) is relatively compact iff S n A is compact.]

1.2.52 Exercise. There exist relatively compact sets that are not compact .


[Hint: The set S �f (0.2, 0.3) is a relatively compact subset of the subset A � (0. 1 , 0.4) of ffi..]

1.3. Baire Category Arguments

1.3.1 DEFINI TION. A SUBSE T S OF A TOPOLOGICAL SPACE X IS : a) nowhere dense IFF EACH NEIGHBORHOOD CON TAINS A NEIGHBORHOOD THAT DOES NOT MEET S; b) OF THE first (Baire) category IFF S IS THE UNION OF COUNTABLY MANY NOWHERE DENSE SETS; c) OF THE second (Baire) category IF S IS NOT OF THE FIRST CATEGORY.

1.3.2 Exercise. A union of finitely many nowhere dense sets is nowhere dense.

1 .3.3 Exercise. A subset S of a topological space X is nowhere dense iff X \ S = X.

1.3.4 THEOREM. A COMPLETE METRIC SPACE (X, d) IS OF THE SECOND CATEGORY.

PROOF. If X = U An and each An is nowhere dense, for any N (x) , by nEN

induction there are sequences {xn }nEN and {N (xn ) }nEN such that: a) {N (Xn ) } C N(x); b) for each n, nEN

sup { d(y, z) :

{Xn} nEN is a Cauchy sequence. Since X is complete, W �f lim Xn ex-n-->(X) ists. On the one hand, for some no, W E Ano and on the other hand, w E N (xno+d e N (xno) , a contradiction since N (xno) n Ano = 0. 0 1.3.5 Example. a) In ffi., Q is a set of the first category and IT �f ffi. \ Q is of the second category. b) For an enumeration {qn } nEN of Q, if 2 � m E N, the complement Nm in ffi. of the set Sm �f U (qn - m -n , qn + m -n) is

nEN nowhere dense. Thus A �f U Nm is of the first category and 1.3.4 implies

mEN

Section 1 .3. Baire Category Arguments 19

B �f ffi. \ A = n Sm is of the second category. If E > 0, B is contained in mEN

a countable union of open intervals and the sum of their lengths is less than E.

1.3.6 THEOREM. (Baire's Category Theorem) IF (X, d) IS A COMPLETE METRIC SPACE AND {Un}nEN IS A SEQUENCE OF DENSE OPEN SUBSETS OF X, THEN U �f n Un IS DENSE IN X.

nEN

PROOF. If each Un is replaced by Vn �f U1 n . . . n Un , each Vn is open and dense, n Vn = n Un �f U, and, to boot, Vn ::) Vn+1 • Thus it may be

nEN nEN assumed at the outset that Un ::) Un+ 1 • Since each Un is dense, if V is a nonempty open subset of X, induction leads to a sequence {xn } nEN in X and a sequence {En}nEN such that: a) 0 < El < 1, B (Xl , E I ) C UI n v; b) if n > l , 0 < En < 2-n , and B (Xn , En ) C B (xn- l , En- I )° n Un . Then {xn} nEN is a Cauchy sequence and, since X is complete, there is an x such that lim d (x, xn ) = o. Furthermore, n-+=

nEN nEN

whence x E V n U: U meets every nonempty open set V. o

1.3.7 THEOREM. IF (X, d) AND (Y, J) ARE METRIC SPACES, T E yX , AND T(X) = Y, T IS OPEN IFF FOR EACH Y IN Y, WHENEVER A SEQUENCE {Yn }nEN IN Y CONVERGES TO Y AND Y = T(x) , FOR SOME SEQUENCE {Xn}nEN CONVERGING TO x, T (Xn ) = Yn , n E N.

PROOF. If T is open, Yn ---+ Y as n ---+ 00, T(x) = Y, and

clef { 1 } Nk(X) = � : d(� , x) < k '

then T [Nk (X)] �f Uk is open and Y E Uk. For some least nk , if n 2': nk , then Yn E rh. By definition, Nk (X) contains a �n such that T (�n ) = Yn . Since Uk ::) Uk+l , nk :::; nk+l . If nk < n < nk+ l , then Yn E Uk . Hence T- 1 (Yn ) n Nk (x) -j. 0. Thus the formula

{ an element of T- 1 (Yn ) if 1 :::; n < nl Xn = �n if n = nk

an element of T- 1 (Yn ) n Nk (X) if nk < n < nk+l

defines a sequence {xn }nEN such that T (xn ) = Yn , n E N. Furthermore if k EN and n 2': nk , then Xn E Nk (x), whence Xn ---+ x as n ---+ 00.


If T is not open and Y E Y, for some x in X, Y �f T(x) , while for some N(x) and each n in N, T[N(x)]1; Nn(y) �f { 1] : 15(1], y) < � } , i.e. , in Nn (y) there is a Yn not in T[N(x) ] . Hence Yn -+ Y as n -+ 00, but if T (xn) = Yn , then Xn tJ- N(x) , i.e. , some sequence {Yn} nEN converges to Y and no sequence {xn} nEN such that T (xn ) = Yn , n EN, converges to x.

o

1.4. Homotopy, Simplices, Fixed Points

The results in the current Section are widely applicable in many parts of mathematics. The main result is Brouwer's Fixed Point Theorem. The machinery that leads to a proof is important in and of itself.

• Homotopy plays a central role in Section 5.5 where the relation of holomorphy and homotopy is explored.

• Simplices or rectangular versions of them occur in Section 4.6 in the treatment of differentiation, in Section 5.3 where the fundamental theorems and formul& of Cauchy are discussed, and in Section 7.1 to provide a direct approach to Runge's Theorem.

• In Section 4.7, Brouwer's Fixed Point Theorem appears to be essential in the derivation of the formula for change of variables in multidimensional integration. More generally, Brouwer's Fixed Point Theorem is central for many of the developments in general and algebraic topology, e.g. , dimension theory, antipodal point theory, etc. [HW, Kel, Sp, Thc].

1.4.1 DEFINITION. FOR TOPOLOGICAL SPACES X AND Y, A SUBSET A OF X, AND SOME F IN C(X X [0, 1] ' Y) THE MAPS h AND r5} IN C(X, Y) ARE F -homotopic in A IFF FOR EACH (x, s) IN X X [0, 1] ' F(x, s) E A AND

{x E X} '* {{F(x, O) = 'Y(x )} 1\ {F(x, 1) = r5(x) } } . (1 .4.2)

THE MAP F IS A homotopy in A OF 'Y INTO 15 ; THE CIRCUMSTANCE DESCRIBED IS SYMBOLIZED 'Y "'F,A 15, 'Y "' F 15, OR 'Y '" 15.

1.4.3 Exercise. If 'Y '" F,A 15 and 15 "'G,A 1], 'Y "'GoF,A 1], i.e. , 'Y '" 15 symbolizing that for some F, 'Y '" F,A 15, '" is an equivalence relation among the elements of C(X, Y) .

The homotopy equivalence class of a curve 'Y is h}.

1.4.4 Example. If 'Y (t) = 3e27rit , 15 (t) = e27rit and F (t, 8 ) �f (3 - 28 ) e27rit , then 'Y '" F,e 15.

Section 1.4. Homotopy, Simplices, Fixed Points 21

For x �f (XO," " xn ) in ffi.n+l , d as in 1.2 . 10, and a positive r, the sets

Bn+l (x,r) �f { y : d(x, y):Sr } (a closed (n + I)-ball) , Bn+l (x,rt �f { y : d(x, y)<r } (an open n + I-ball) , and

sn(x,r) �f { y : d(x, y) =r } (an n-sphere)

are related according to

In particular,Bn+1 �f Bn+1(O, 1) is the n + I-ball and is homeomorphic to every n + I-ball. Similarly, sn �f sn(o, 1) is the n-sphere and is homeomorphic to every n-sphere.

1.4.5 Exercise. a) The n + I-cell, [0, It+1 is homeomorphic to Bn+l . b) The boundary 8[0, I]n+l is homeomorphic to sn .

The object of the subsequent discussion is to prove that the identity map id : sn r-+ sn and any consta.nt map c : sn r-+ sn for which c (sn) is a single vector in sn are not homotopic in sn. The technique involves the decomposition of sn into small pieces, the spherical n-simplices described below.

1.4.6 DEFINITION. FOR A k + I-TUPLE {xo , xl , . . . , xd OF VECTORS IN

IDln+ 1 • <!<:f { � \ . . \ . . k � \ - 1 } m. , THE SET a - � AJXJ • AJ > 0, ° :S J:S , � AJ - , DE-J=o J=o

NOTED (Xo, ... , Xk) IS A k-simplex IFF {Xo , . . . , Xk} IS linearly independent . OTHERWISE, a IS A degenerate k-simplex . THE closed k-simplex resp. closed degenerate k-simplex IS

{ k _ clef an = L AjXj

J=O

DENOTED [xo , . . . , Xk]. FOR A SUBSET { io , . . . , ip} OF {O , I , . . . , k} , THE CORRESPONDING p

face OF a IS

A O-FACE IS A vertex; A I-FACE IS AN edge .


[ 1.4.7 Note. Every face of an resp. an is a relatively open subset of an resp. an. Thus an edge is homeomorphic to (0, 1 ) ; a 2-face is homeomorphic to B2 (0, lr , etc. ]

1.4.8 Exercise. If a in ffi.Tn is a k-simplex: a) m 2: k; b) a is an open subset of the hyperplane Ha �f Xo + span {Xl - Xo , . . . , Xk - xo} in the topology induced on H a by ffi.Tn; c) each vector in a belongs to one and only one face of a; d) a is a closed subset of ffi.Tn and is homeomorphic to Bk. A (possibly degenerate) simplex is a convex set.

When x -j. 0, p(x) � 1 I�1 2

E sn is the mdial projection of x onto sn. 1.4.9 Exercise. If {xo , . . . , xn } C sn, d (xi , xj) < 1 , ° � i , j � n, and

( ) clef X E Xo , . . . , Xn = a, then x -j. 0. (The set p (a) is the spherical simplex (uniquely) determined by an . Its faces are the radial projections of the faces of an . ) n

[Hint: If L ti = 1 and ti 2: 0, ° � i � n, then i=O

The inequality

n n L tiXi = Lti (Xi - Xo ) + Xo· i=O i=O

I IX -YI 1 2 2: I I Ix I 1 2 - I IY I 1 2 1 applies (v. 3.1 .2 and (3.2. 12) ) . ] When sn is the union of nondegenerate spherical n-simplices and any

n - I-face is in the intersection of precisely two spherical n-simplices, the spherical n-simplices constitute a triangulation of sn . 1.4.10 Exercise. When 1 � i � n + 1 , the vectors

°

ei �f 1 � ith component

°

determine 2n+1 nondegenerate n-simplices (±el , . . . , ±en+l ) . Their radial projections constitute a triangulation of sn . 1.4. 11 Exercise. The diameter of the n-simplex a �f (xo , . . . , xn) is

Section 1 .4. Homotopy, Simplices, Fixed Points 23

Thus diam (a) is the maximum of the lengths of the (finitely many) edges of a, whence diam (a) is the length of some edge of a.

1.4. 12 DEFINITION. THE barycenter OF THE n-SIMPLEX

clef ( ) a = Xo , ... , Xn

clef 1 � IS THE VECTOR b(a) = -- � Xk. n + 1 FOR THE PARTIAL ORDER -< DE-k=o

FINED AMONG THE FACES OF a BY

THE SIMPLICES OF THE barycentric subdivision OF a ARE THOSE AND ONLY THOSE DETERMINED BY BARYCENTERS {b (ap,) , ... , b (apr)} SUCH THAT ap1 -< ap2 ... -< apr. THE UNION OF THE SIMPLICES OF THE BARYCENTRIC SUBDIVISION OF a IS DENOTED a'. M ORE GENERALLY, aU) DENOTES THE UNION OF n-SIMPLICES OF THE BARYCENTRIC SUBDIVISION OF ALL THE SIMPLICES IN aU-I), 2 :::; j.

1.4.13 LEMMA. IF a IS AN n-SIMPLEX AND 7 IS AN n-SIMPLEX IN a', n THEN diam (7) :::; --diam (a). n + l

PROOF. If b (ap) and b (aq) are vertices of 7 and ap -< aq, via appropriate labelling, ap = (xo, ... , xp) and aq = (xo, ... , xp, Xp+I, . .. , xq). Direct calculation leads to the equation

( p q ) q - p 1 1

b(ap) - b(aq) = --1 --1 LXk - -- L Xk q + p + k=O q - P k=p+ I

Since --LXk,-- L Xk {I p 1 q } P + 1 k=O q - P k=p+l

1.4.14 COROLLARY. IF a IS AN n-SIMPLEX AND 7 IS AN n-SIMPLEX IN aU), diam (7):::; (n: 1r diam (a).

PROOF. Mathematical induction, based on the conclusion of 1.4.13 applies. o


1 .4.15 Exercise. If E > 0, sn admits a triangulation T such that the diameter of each simplex of T does not exceed E. (The triangulation T is of mesh E. )

If F is a homotopy in sn of id to c and T is a triangulation of sn , to each s and each vertex x of T there corresponds a vector F (x, s) �f Xs of sn . Because sn x [0, 1 ] is compact, F is uniformly continuous (v. Sec-tion 1 .6) . Hence, if the mesh of T is small, and a �f (xo , . . . , xn ) is a spherical n + I-simplex in T, for all s in [0, 1 ] ' the diameter of the (possibly degenerate) simplex aF(.,s) �f (F (xo , s) , ... , F (xn , s ) ) is also small.

[ 1 .4.16 Note. For fixed s , the set TF(.,s) �f {aF("S) } of aET

spherical simplices does not necessarily constitute a triangulation of sn. The union of the constituent spherical simplices need not cover sn; the intersection of two or more of them can have a nonempty interior relative to Sn-they can overlap.]

When x E Sn and x is not on the boundary of any aFt-,s), the cardinality of the set of spherical n-simplices aF(.,s) to which x belongs is denoted N (x, F (·, s ) , T). 1.4.17 Exercise. a) If Sf is near s, then

N (x, F (· , s ) , T) = N (x, F ( . , Sf) , T).

b) When x remains fixed and s traverses [0, 1 ] ' the number N (x, F (·, s ) , T) remains constant . c) Where N (·, F (·, 0) , T) is defined, N (·, F (·, 0) , T) = 1 .

[Hint: The function N (·, F (·, s ) , T) is continuous and N+ -valued. ]

1 .4.18 Exercise. If T is a triangulation of sn and F is any map of the vertices of T into TF in the manner described above (F is a vertex map) , there is a vertex map Ff such that: a ) for TF', each spherical n-simplex is determined by linearly independent vectors; b) if x is on no boundary found in TF, x is on no boundary found in TF' and N (x, F, T) = N (x, Ff, T).

[Hint: A hyperplane in ffi.n+l is nowhere dense. ]

1.4.19 Exercise. If neither x nor y is on any boundary found in TF and the vectors determining each spherical n-simplex of TF are linearly independent , there is a path 1f connecting x and y on sn , meeting each n - I-face of TF at most finitely often. If k > 1, 1f meets no n - k-face.

[Hint: The union of the spherical n-simplices determined by TF is connected. The hypothesis of linear independence implies that an escape from one spherical n-simplex to another can always be managed through an n - I-face. There are only finitely many spherical n-simplices in TF.]


1.4.20 Exercise. At each crossing of 1f from one spherical n-simplex to another, the number N( · , F, T) changes by ±2 or o.

[Hint: If the adjoining n-simplices overlap, crossing through their common n - I-face involves either entering or leaving both. If the adjoining n-simplices do not overlap, the number N( · , F, T) does not change. ]

1.4.21 Exercise. For a given triangulation T and a vertex map F such that the spherical n-simplices of TF are determined by linearly independent vectors, the number N(x, F, T) is, modulo 2, independent of x.

In sum, the vertex map F and the triangulation T determine a number N (F, T) such that for all x in sn ,

N(x, F, T) == N(F, T) (mod 2 ) .

Since, modulo 2 , N( · , F( . , 0 ) , T) = 1 and N( · , F( · , 1 ) , T) = 0, the homotopy F does not exist: id and c are not homotopic. [ 1.4.22 Remark. When J E C (sn , sn ) and T is a triangulation of sn , J confined to the vertices of T is a vertex map Ff . The number N (Ff , T) (mod 2) is the degree modulo 2 of the pair {J, T}. In fact, N (Ff , T) does not depend on T and is an intrinsic property of the pair {J, sn} .

I f orientation of simplices is taken into account , a more general Brouwer degree (taking values in Z) of the map J can be defined. The Brouwer degree, which may be viewed as counting the number (positive or negative) of times J wraps sn about itself, is a fundamental and important topological invariant of the pair {J, sn} .

For example, i f n E Z, the map

n : S1 3 (cos B, sin B) r-+ (cos nB, sin nB) E S1

wraps S1 about itself n times.

Among other applications of the Brouwer degree is a proof of the Fundamental Theorem of Algebra (FTA) [Du, DuG, Sp] . ]

1.4.23 Exercise. There is no map g : Bn+1 r-+ Sn such that g lsn = id . [Hint: The map F : sn x [0, 1 ] 3 (x, s ) r-+ g[(1 - s)x ] is a forbidden homotopy. ] Brouwer's Fixed Point Theorem says that a continuous map J of Bn+1

into itself leaves some point x fixed: J(x) = x.


1 .4.24 Exercise. If g E C([-I , 1], [-1 , 1]) , for some x in [ - 1 , 1]' g(x) = x,

i.e. , Brouwer's Fixed Point Theorem is valid when n = o. On the other hand, slight modifications of the hypothesis invalidate

the conclusion. 1.4.25 Exercise. The map

maps B2(O, 1 ) ° onto itself and leaves no z in B2 (O, It fixed. 1 .4.26 Exercise. The map

lZ f : (D(O, 1) \ {O}) :1 Z r-+ � maps D(O, 1) \ {O} onto itself and leaves no z fixed.

Here is an intuitive argument for Brouwer's Fixed Point Theorem. If the conclusion is false, for some continuous f and each x in Bn+l ,

f (x) -j. x. The half-line starting at f (x) and through x meets sn in a point g( x) . The map g is continuous and g I Sn = id . Intuition suggests g does some tearing, i.e. , g is not continuous.

As the following material reveals , via the technology developed above, the intuitive argument just offered can be made rigorous.

1 .4.27 THEOREM. (Brouwer's Fixed Point Theorem) IF f : Bn+1 r-+ Bn+1 IS CONTINUOUS, FOR SOME x IN Bn+l , f (x) = x.

PROOF. Otherwise, for each x, f (x) -j. x and the line

{ f(x) + t [x - f (x)] : t � O } meets sn in a vector g(x) . The map g : Bn+1 r-+ Bn+1 is continuous and if x E sn , then g(x) = x: g l sn= id , a contradiction of 1 .4.23. 0 1 .4.28 Exercise. If X and Y are homeomorphic and each f in C(X, X) leaves some x in X fixed, each g in C(Y, Y) leaves some y in Y fixed.

The fixed point property is a topological invariant An important aspect of the development above is that for the purpose

in hand, discussions of continuous maps may be reduced to discussions of maps of finite sets of points, e.g. , the vertices of the triangulations T. A significant block of topology is devoted to a finitistic approach-simplicial approximation. In this direction, the work, e.g. , of Alexander, tech, Eilenberg, Hurewicz, Lefschetz, Massey, Mayer, Spanier, Steenrod, and Vietoris,


gave rise to modern algebraic topology [Bro, Du, DuG, Sp] ; v. also [F, The].

In the discussion below, a similar finitistic analysis provides an alternative derivation of Brouwer's Fixed Point Theorem.

clef 1 .4.29 LEMMA. (Sperner) IF an = (xo, ... , Xn) IS AN n-SIMPLEX AND

IS SUCH THAT FOR EACH p-FACE ap OF an, W [b (ap)] IS A VERTEX OF ap,

i.e., IF W IS A Sperner map , THEN FOR SOME Tn �f (Yo, ... , Yn) OF a�,

(FOR SOME PERMUTATION 1f OF {O, ... ,n}, w (Yd = X7r(k) , O<::;k<::;n -)

[ 1.4.30 Note. Sperner 's Lemma is purely combinatorial. No topological or algebraic properties, e.g. , continuity, linearity, are assumed for w.]

PROOF. When n = 0 the conclusion is automatic since w = id . If the result obtains when n = m - 1 , a relabeling of the vertices permits the assumption that w [b (am)] = Xm. Furthermore, for w confined to the simplex am-I �f (xo, ... , xm-d , Sperner's Lemma obtains. Hence there is an (m - I )-simplex Tm-I �f (Yo, ... ,Ym-d contained in a:n-I and such that {w (Yk)}:'=-OI = {Xd:,�I. However Tm-I is the face of some m-simplex Tm in a:". The vertices of Tm are the vertices of Tm- I together with b (am). Since w [b (am)] = Xm , W maps the vertices of T m onto the vertices of am.

o 1.4.31 Exercise. If n = 1 , barycentric subdivision is simply bisection and Sperner's Lemma is a triviality.

Since each vertex of a�k) is a barycenter b (a�,k-I)) of some a�k-I) , a

Sperner map w . a rk) r-+ a (k-I) carries b (a (k-I)) onto a vertex of a (k-I) n · n n n n ,

which is a barycenter b (a�k-2)) of some a�k-2). Hence a Sperner map

Wn-I : a�k-I) r-+ a�k-2) carries b (a�k-I)) onto a vertex of a�k-2). The composition Wn-I 0 Wn : a�k) r-+ a�k-2) is a Sperner map. By induction, there is a Sperner map w(k) �f WI 0 .. · 0 Wn : a�k) r-+ an-


1 1

2 3 2

Figure 1 .4. 1 .

1.4.32 Example. The marking of the vertices in Figure 1.4.1 is quite arbitrary.

1 .4.33 Exercise. The counterpart of Sperner's Lemma obtains for w(k).

[Hint: For a I-simplex, the result is an instructive consequence of simple counting and induction. For the general case, induction applies. ]

In Table 1.4.1 there are two examples Wi : O"� r-+ 0"2 , i = 1 , 2 , of Sperner maps of O"� into 0"2 . Only the actions of the linear maps Wi on the vertices are described. If x E O"� , x is in some unique 0-, 1-, or 2-simplex 7 of O"� . Hence x is a convex (linear) combination of the vertices of 7.

WI w2

1 � 1 1 � 1 2 � 2 2 � 2 3 � 3 3 � 3 P � 2 P � I S � 3 S � I R � 2 R � 2 Q � 3 Q � 2

72 : ( 1 , P, Q ) (3, Q, S)

Table 1 .4.1


Correspondingly, Wi (X) is assumed to be the same convex combination of the wi-images of those vertices.

In each instance, T2 is a Sperner simplex , i.e. , a simplex behaving as stated in Sperner's Lemma.

Below the description of each map, a corresponding Sperner simplex T2 is indicated by its vertices.

[ 1 .4.34 Note. As the calculations for the two maps WI and W2 reveal, the respective associated Sperner simplices are different. However, for each map, further calculations show that each T2 is the only Sperner simplex: all the remaining five simplices fail to satisfy the defining conditions for a Sperner simplex. In general, the Tn in Sperner's Lemma is unique. The proof of uniqueness is omitted since the uniqueness of Tn is not essential for the work that follows. ]

The penultimate step in the PROOF of Brouwer's Fixed Point Theorem is the next consequence of Sperner's Lemma. Both the statement and PROOF of 1 .4.35 display essential links between the combinatorics in Sperner 's Lemma and the underlying topological structure of the context. The combinatorial aspects may be termed finitistic-various technical counting procedures are involved-but no appeal is made to continuity or related topological notions. The topological arguments deal with the metric properties of covers of a simplex. The essential fact proved in 1.4.35 is the following:

If a cover consisting of closed sets is sufficiently redundant, the intersection of all elements of the cover is nonempty.

The statement above is a rather elaborate extension of the pigeon-hole principle:

If n objects are placed in fewer than n pigeon-holes, at least one pigeon-hole contains two objects.

The argument relies on the metric properties of coverings.

L I clef ) clef {F }n 1.4.35 EMMA . F an = (xo , . . . , xn IS AN n-SIMPLEX, :F = k k=o IS A SET OF CLOSED SUBSETS OF (j n , AND FOR EACH SUBSET {io , ... , ip} OF

p n {O, . . . , n} , (XiO > " . . ' Xip ) C U Fik , THEN n Fk -j. 0.

k=O k=O


n PROOF. By hypothesis, Xi E Fi, 0 :::; i :::; n, an C U Fi , and every vertex of

i= l n a barycentric subdivision is in some Fi· If n Fk = 0, then

k=O

is an open cover of an . From 1 .4.14 it follows that for some r, the maximum diameter of any simplex in at) is less than the Lebesgue number l1(U) (v.1 .2.36- 1.2 .40) . Each Y in (at) ) is in some Fi, which contains Xi.

For some Sperner map w, the map at) 3 Y r-+ xi is w(r) . Sperner's L . 1· h t f . 1 clef ( ) . (r) emma Imp les t a or some n-slmp ex Tn = Yo , ... , Yn In an ,

The vertices Yi (relabeled as needed) may be assumed to be such that Yi E Fi, 0 :::; i :::; n. Thus Tn n Fi -j. 0, 0 :::; i :::; n, and so diam (Tn ) 2': l1(U) , a contradiction. 0

The trapezoids F1, F2 , and F3 in Figure 1 .4.2 exemplify the closed 3

sets described in the LEMMA: n Fi is the shaded triangle. i= l

1

2 D B 3 Fl = lCD2, F2 = 2EF 3, F3 = lAB3

Figure 1 .4.2.


1.4.36 THEOREM. (Brouwer 's Fixed Point Theorem, again) IF X AND Bn

ARE HOMEOMORPHIC AND f E C(X, X ) , SOME x IN X IS A FIXED POINT FOR f: f(x) = x.

PROOF. Owing to 1 .4.8d) and 1.4.28, the truth of the result when X is the closed simplex

- clef [ clef ( ) clef ( ) clef ( ) ] an = eo = 0, 0, . . . , 0 ,el = 1 , 0, . . . , 0 , . . . , en = 0, 0, . . . , 1

implies the result for any X as described. If X = an and x E X, there are unique nonnegative Ak (X), 1 ::; k ::; n, such that

n

n and x = L Ak(x)ek. Then 1 .2 .15 implies that the functions

k=O

are continuous. Since f (an) C an, if x E an and

n n f(x) = L Ak [J(x)] ek �f L I-lk (x)ek,

k=O k=O

the functions I-lk, ° ::; k ::; n, are continuous. Hence the sets

are closed. Furthermore, if {io , . . . , ip} C {O, . . . , n},

p ( eio , ... , eip ) C U Fik •

k=O

n Thus, according to 1.4.35, F �f n Fk -j. 0.

k=O If x E F , then I-lk (X) ::; Ak(X), 0 ::; k ::; n. Since

n n 1 = L I-lk (X) = L Ak(X),

k=O k=O

I-lk(X) = Ak(X), 0 ::; k ::; n, i.e. , f (x) = x. o


1.5. Appendix 1 . Filters

1.5.1 DEFINITION. FOR A SET X, A filter F �f {F} IN !fj(X) IS A SUBSET CONFORMING TO a)-c) OF 1 .2.8. A FILTER F' refines OR IS A refinement OF OR IS finer THAN A FILTER F IFF F' ::) F IN WHICH CASE ONE WRITES F' >- F. WHEN F' IS FINER THAN F, F IS coarser THAN F' .

The set of all filters in !fj( X) is partially ordered by >- and for a given filter F there is, by virtue of any one of the equivalents-Hausdorff Maximality Principle, Axiom of Choice, Well-ordering Axiom, etc . , [Mel-of Zorn's Lemma, a maximal filter U such that U >- F. A maximal filter U is an ultrafilter.

The filter consisting of the single set X is coarser than any filter. For x in X, the filter consisting of all the supersets of {x} is an ultrafilter.

1.5.2 DEFINITION. A SUBSET 8 OF !fj(X) IS A FILTER BASE IFF: a) 8 -j. 0; b) 0 1- 8; c) THE INTERSECTION OF ANY TWO ELEMENTS OF 8 CONTAINS AN EL

EMENT OF 8.

1.5.3 Exercise. The set of all supersets of a filter base 8 is a filter F.

1.5.4 Exercise. The intersection of all filters containing a filter base 8 is a filter F, the filter generated by 8. 1.5.5 Exercise. Every filter F is a filter base; F generates itself. 1 .5.6 Exercise. For a map f : X r-+ Y: a) If 8 is a filter base in !fj(X) , {f(B) }BEB �f f(8) is a filter base in !fj(Y) . b) I f C is a filter base in !fj(Y) , {I-I (C) } CEe �f f- 1 (C) is a filter base in !fj( X) iff for all C, {I-I (C) -j. 0} .

When X is a topological space, a filter F converges to x in X iff every N(x) contains an element of F: F invades every N(x). Since N(x) is itself a filter, F converges to x iff F refines N(x) . When F converges to x, x is the limit of :F.

A filter F � {F} in !fj( X) is a diset with respect to the partial order induced by reversed inclusion among the elements of F: F' >- F iff F' c F. A map n : F 3 F r-+ n( F) E F (c X) is thus a net corresponding to the filter :F. Conversely, when A is a diset, n : A 3 A r-+ n(A) E X is a net, and B>. �f { n(p,) : p, >- A }, {B>.hEA is a filter base that generates the filter Fn corresponding to the net n. The correspondence net ---+ filter is an injective map. The correspondence filter ---+ net is not a map since there can be more than one net corresponding to a given filter.

[ 1.5.7 Note. Owing to the correspondences between nets and filters , any notion formulable in terms of nets is equally formulable in terms of filters. ]

Section 1 .5 . Appendix 1. Filters 33

1.5.8 Exercise. A filter F converges to x iff every net n corresponding to F converges to x.

1 .5.9 Exercise. If the filter F corresponds to the net n, then n converges to x iff F converges to x.

1.5.10 THEOREM. A TOPOLOGICAL SPACE X IS COMPACT: a) IFF EVERY ULTRAFILTER IN !fj(X) CONVERGES ; b) IFF FOR EVERY NET n : A r-+ X THERE IS A CONVERGENT COFINAL NET n lA' .

PROOF. a) If X is compact and U �f {F} is an ultrafilter in !fj(X) , each F is closed. If n F = 0, then for some finite subset {F1 , . . . , Fn } of U, n n

FEU

n Fk c n Fk = 0, which is impossible for elements of a filter. Hence k=l k= l some x is in n F. For the filter N(x) ,

FEU

{ N (x) n F : N (x) E N(x) , F E U }

generates a filter V that converges to x. Furthermore, V >- U and, since U is an ultrafilter , V = U.

Conversely, if every ultrafilter converges , and E �f {E} is an fip set of closed sets, then E generates a filter F, contained in an ultrafilter U that converges, say to x. For N (x) and Eo in E there is in U a U that is contained in N (x) and N ( x ) ::) U n Eo -j. 0. Since Eo is closed, x E Eo: x E n E. In other words, if a set E of closed sets is an fip set (v. 1 .2.25) ,

EE£ the intersection of all the sets of E is nonempty. Hence X is compact.

b) If X is compact and n : A r-+ X is a net, the corresponding filter F �f {B>. �f { n(p,) : p, >- A }} is a subset of an ultrafilter U converging to some x in X. Hence x is in the closure of each element U of U, in particular x is in the closure of each B>. . Thus, for each N (x) and each A in A, there is a U contained in N (x) . In the nonempty set B>. n U there is an n (N) and A' >- A. The set A' of all such N is cofinal in A and the co final net n I A' converges to x.

Conversely, if U is an ultrafilter in !fj(X) and n is a net corresponding to U, by hypothesis there is a cofinal net converging to some x in X. Hence U converges to x, whence every ultrafilter converges and X is compact.

o The following proof of Tychonov's Theorem (1 .2 .30) is phrased m

terms of filters. PROOF. For A in A, the map 1f>. : X 3 x �f {xJL}JLEA r-+ x>. E X>. is continuous . If U is an ultrafilter in X, 1f>. (U) is a filter base of a filter U>. in


Xx . Since 7rA1 (U).. ) contains U, U).. is itself an ultrafilter. Because X).. is compact, U).. converges to some x).. in X).. . The filter U converges to the . clef { } . X 0 pomt X = X).. )..EA m .

1.6. Appendix 2. Uniformity

For a metric space (X, d) there is in lfj (X2) an associated filter base

B �f {B_ �f { (X, y) : (x, y) E X2 , d(x , y) < E } } . _>0 The operations

o :lfj (X2) x lfj (X2) :1 (A, B) , r-+ { (x, y) : 3(z){ ((x, z) E A) 1\ ((z, y) E B) } } , �f A 0 B E lfj (X2) , and

-1 :lfj (X2) :1 A r-+ { (x, y) : (y, x) E A } �f A-I permit a succinct description of two fundamental properties of the filter U generated by B, viz. :

a) If U E U, then U- I E U; b) If U E U, for some V in U, V 0 V-I c U.

For the particular filter U associated with the metric d it is true also that

c) n U = { (x, x) : x E X } �f 6. UEU

The preceding discussion motivates the following

1 .6.1 DEFINITION. FOR A SET X, A FILTER U IN lfj (X2) IS A uniformity IFF a) AND b) OBTAIN . IF c) OBTAINS AS WELL, U IS A Hausdorff uniformity .

A PAIR (X,U) CONSISTING OF A SET X AND A UNIFORMITY U IS A UNIFORM SPACE, USUALLY DENOTED SIMPLY X. EACH ELEMENT OF U IS A VICINITY.

Uniform spaces are discussed in [Bou, Tuk, We1 ] .

1.6.2 Example. When G is a topological group (v. Section 4.9) and N(e) is the filter of neighborhoods of the identity e of G, the set

U �f {UN �f { (x , x') : XX,- I E N } } NEN(e)

Section 1.6. Appendix 2. Uniformity 35

is a uniformity for G and (G, U) is a uniform space.

1.6.3 DEFINITION. FOR A UNIFORM SPACE (X, U) WHEN x E X AND U E U THE U-INDUCED NEIGHBORHOOD OF x IS

Ux �f { y : (x, y) E U } .

1.6.4 Exercise. The system {UX} XEX is a base of neighborhoods of a UEU

topological space (X, T u ) . When (X,U) and (Y, V) are uniform spaces, a function f : X r-+ Y is

uniformly continuous iff for each V in V there is in U a U such that

{ (x, y) E U} '* { (f(x) , f(y) ) E V} .

A net n : A r-+ X is a Cauchy net iff for each U in U there is in A a Ao such that {P >- Ao} 1\ {I.l >- Ao} } '* ( (n(A) , n(I.l) ) E U}. A net n : A r-+ eX is a uniform Cauchy net iff for every positive E there is in A a Ao such that

Vx {{A >- Ao} 1\ {I.l >- Ao}} '* { In(A) (x) - n(l.l) (x) I < E} .

1.6.5 Exercise. If (X, U) and (Y, V) are uniform spaces and f : X r-+ Y is uniformly continuous, f is (T u , Tv )-continuous. 1.6.6 Exercise. If (X, U) is a uniform space compact in the U-induced topology, (Y, V) is a uniform space for which V is a Hausdorff uniformity, and f E yX is continuous with respect to the uniformity induced topologies T u and Tv, each of the following obtains . a) The map

F : X x X 3 (a, b) r-+ [J(a) , f(b) ] E Y x Y

is continuous . b) If W E y, then W is open in Y x Y, V �f F -1 (W) is open in X x X, and the diagonal 6 in X x X is contained in V: 6 C V. c) If V is not in U:

clef { - } • :F = (U \ V) UEU is an fip set of closed subsets of the compact space

X x X (whence A �f n F i- 0) ; FEF

• the diagonal 6 = n u, is compact , and 6 n ( n F ) = 0; UEU FEF

• if (x, y) E A, each neighborhood of (x, y) meets 6 (whence (x, y) E 6) and (x , y ) E (X x X \ 6) , a contradiction: V E U.

d) The map f is uniformly continuous.


A continuous map of a compact uniform space into a uniform Hausdorff space is uniformly continuous.

1 .6.7 THEOREM. IF: a) (X,U) AND (Y, V) ARE UNIFORM SPACES , AND T U AND Tv ARE THEIR UNIFORMITY INDUCED TOPOLOGIES , b)

n : A 3 A r-+ I>. E C(X, Y)

IS A NET, AND c) f E yX , AND FOR EACH VICINITY V IN V, THERE IS A Ao SUCH THAT {>. >- Ao } ::::} {Vx { [J.x (x) , f(x)] E V}} , THEN f E C(X, Y) .

The uniform limit of continuous functions is continuous.

PROOF. If x E X, Y = f(x) and N(y) is a neighborhood derived from a vicinity V in V, there is a vicinity Z such that Z-l 0 Z-l C V and there is a vicinity W such that W o W C Z-l . For some Ao , if A � Ao , then VxVx' { [I>. (x) , f (x)] E W} 1\ { [I>. (x') , f (x')] E Z}, and for some N(x) derived from a vicinity U, {x' E N(x) } ::::} { [I>.o (x), 1>.0 (x' )] E W}. Since W o W 0 Z-l C Z-l 0 Z-l C V, it follows that [J(x) , f (x' )] E V. 0

For uniform spaces (X, U) resp. (Y, V) and their uniformity-induced topologies T u and Tv, a set S �f { I>. LEA of elements in C(X, Y) is equicontinuous iff for each vicinity V of V there is in U a vicinity U such that if (x, x' ) E U, for all A, [h(x) , h (x')] E V.

1 .6.8 THEOREM. IF (X, U) RESP. (Y, V) ARE UNIFORM SPACES THAT ARE COMPACT HAUSDORFF SPACES IN THEIR UNIFORMITY-INDUCED TOPOLOGIES AND THE NET n : A 3 A r-+ I>. E C (X, Y) IS AN EQUlCONTINUOUS SET OF FUNCTIONS, SOME COFINAL SUBNET n : A' 3 >.' r-+ hI CONVERGES UNIFORMLY TO AN f (NECESSARILY IN C(X, Y) ) .

PROOF. For each x in X if Yx �f Y , then XY = Xx EX Yx and 1.2.30 implies Y x is compact. Consequently, 1.5 .10 implies there is a cofinal subnet n : A' 3 A' r-+ fA' that converges to some f in yX .

If V is a vicinity in V, there is a vicinity Z such that Z-l 0 Z-l C V and there is a vicinity W such that W o W C Z-l . The equicontinuity of the functions in {fA' } A' EA' implies there is in U a vicinity U such that for all >.', { (x, y) E U} ::::} { [JA' (x) , JA' (y)] E W}. The set {NU (x) }xEX of derived neighborhoods is a cover of X and hence, for some finite set {Xm h�m�M'

U Nu (xm) = X. If x E X, for some m, (x , xm ) E U, whence for all l �m�M >.', [JA' (x) , fA' (xm)] E W. By a similar argument, if A' >- A� and 11/ >- f.-l�, [JIL ' (x) , fIL' (xm )] E Z-l . The convergence of

Section 1.6. Appendix 2. Uniformity 37

implies that for some x-Iree (A� , f.1.�) ,

Consequently, if A ' >- A� and f.1.' >- f.1.�, then [Iv (X) , fJLf (x)] E V, i .e. , the convergence is uniform: I v � I· 0

1.6.9 COROLLARY. (Arzela-Ascoli) IF (X, d) IS A COMPACT METRIC SPACE AND {In} nEN IS A UNIFORMLY BOUNDED EQUlCONTINUOUS SEQUENCE IN C(X, JR.) , SOME SUBSEQUENCE {Ink hEN CONVERGES UNIFORMLY ON X TO A g IN C(X, JR.) . PROOF. Although the PROOF of 1 .6.8 applies directly, the following proof uses a diagonalization technique of independent interest.

For each positive r in Q, U B (x , rt = X and since X is compact, xEX

there are finitely many xi (r) , 1 � i � I(r) , such that

X = U B (Xi (r) , r) . l�i�I (r)

Hence {xi (r) }o<rE\Ql, l�i�I (r) is a countable dense subset D �f {Pn }nEN of X.

Since {In } nEN is uniformly bounded, there is a convergent subsequence

a convergent subsubsequence

etc. The diagonal sequence

converges at each point of D. The equicontinuity of {In}nEN implies that if x E X, for a suitable Pk ,

the first and third terms in the right member of

I grn (x) - gn (x) 1 � I grn (x) - grn (Pk ) 1 + I grn (Pk ) - gn (Pk ) 1 + I gn (Pk ) - gn (x) l ,

are small, and then for large m and n, the second term is small: {gn} nEN converges at each x in X, say to g(x).

38 Chapter 1 . Fundamentals

If E > 0, there is a positive 1] such that

Furthermore {Nj �f { x : d (x, pj ) < 1] }} . is an open cover of X and l EN

thus contains a finite sub cover {Nj } l< < J " There is a k (E) such that if _ l _ m, n > k (E) and 1 � j � J, then d [grn (Pj ) , gn (Pj )] < E. Thus, if x E X, for some j in [ 1 , J] , x E Nj and if m, n > k (E) , then

d [grn (x) , gn (x)] � d [grn (x) , grn (pj )] + d [grn (Pj ) , gn (pj )] + d [gn (pj ) , gn (x)] < 3E.

It follows that gn � g, whence g is continuous. 0 [ 1.6.10 Note. There are many variations and extensions of 1 .6.8. Some of these are discussed in [Is, Kel] .

Viewed as a subset of C( X, JR.) topologized by the metric

d : C(X, JR.? 3 (f, g) r-+ sup I f(x) - g(x) l , xEX

{fn } nEN is a precompact set, i .e . , its closure is compact.

The set {fd'xEA resp . {fn}nEN of 1.6.8 resp . 1.6.9 is an example of a normal family F. The general context for a normal family is :

• two sets X and Y, and some topology T for a subset S of yX . ,

• a set F contained in S and such that F is compact (F is T -precompact) .

In several applications of this notion, X and Y are themselves topological spaces and S = C(X, Y) .]

1.7. Miscellaneous Exercises

1. 7. 1 Exercise. If X and Y are topological spaces and f E yX then: a) f is continuous iff for every x in X and every net n : A 3 ), r-+ n(),) E X converging to x, f 0 n converges to f(x) ; b) the filter formulation of a) is valid; c) f is open iff for each x in X there is a neighborhood N (x) such that f[N(x)] is open. What are the filter and net formulations of c)?

Section 1 .7. Miscellaneous Exercises 39

1. 7.2 Exercise. If a set N of subsets of a set X is such that: a) each x of X is in some Ux (an x-neighborhood in N); b) if Ux and Vx are xneighborhoods in N, for some x-neighborhood Wx in N, WX c Ux n Vx ; c) for each x-neighborhood Ux in N and each y in Ux , some y-neighborhood Uy in N is a subset of Ux , then N is a neighborhood base for a topology for X. 1. 7.3 Exercise. If {f, g} c JR.x : a)

fV clef {f } f + g + I f - g l f clef . {f } f + g - I f - g l g = max g = 1\ g = mIn g = . , 2 '

, 2 '

b) {f V g, f 1\ g} C C(X, JR.) iff {f, g} C C(X, JR.) ) ; c) f is continuous iff f is lsc and usc. 1. 7.4 Exercise. If X is a compact space and Y is a topological space, and f : X f-t Y is a continuous bijection, then f- 1 is continuous, i .e . , f is bicontinuous .

1. 7.5 Exercise. Some f in JR.lR is continuous and not open. 1. 7.6 Exercise. If {rn} nEN is an enumeration of Q and

fn(X) �f { l if :Z: E \r1 , . . . , rn} : o otherwIse

a) lim f n (x) �f f (x) exists for each x in JR.; b) each f n is Riemann inte-n -+ =

grable on [0, 1] ; c) f is not Riemann integrable. 1. 7. 7 Exercise. If

if Q 3 x = E , p E Z, q E N, plq = 1 q

otherwise

f is continuous at x iff x tJ. Q. 1. 7.8 Exercise. A point a is a limit point of a set A (a E Ae) iff for each N(a) and each net n : A f-t A \ {a} , n is frequently in N(a) , v. 1.2.22 .

1. 7.9 Exercise. If X and Y are topological spaces, X is connected, and f E C(X, Y) , then f(X) is connected. 1. 7. 10 Exercise. If X is a topological space, C is a connected subset of X, A c X, and C meets both A and X \ A, then C meets 8(A) . 1. 7. 11 Exercise. a) If Q is a region in C and { a, b} c Q, there is a finite set { [Zn , zn+dL<n<N of complex intervals , each contained in Q and such that ZI = a and ZN-= b. Their union is a polygon : Q is polygonally connected. b) A similar result obtains for a region Q in JR.n , n E N.


[Hint: a) The set { z : z is polygonally connected to a } is open, relatively closed in Q, and nonempty.]

1. 7. 12 Exercise. If X is a second countable topological space, X contains a countable dense subset, i .e . , X is separable. 1. 7. 13 Exercise. a) In a topological space X, the closure of a subset A is the intersection of the set of all closed sets containing A. b) If X is a locally compact Hausdorff space and K E K(X), then K is the intersection of the set of all open sets containing K. c) If X is a second countable locally compact Hausdorff space, each K in K(X) is the intersection of a countable set of open sets containing K: each compact set is a compact G/j . d) If X is a metric space, every closed set is the intersection of a countable set of open sets: each closed set is a G/j .

1. 7.14 Exercise. For N regarded as a diset ordered by < and a net

n : N 3 k r-+ n(k ) E !fj(X) ,

x E n resp. x E n iff x is in infinitely many sets n(k) resp. x is in all but finitely many sets n( k) .

1. 7. 15 Exercise. For a topological space X, an a in X, the filter N( a) , and an f in ffi.x , there i s a corresponding net

n : N(a) 3 N(a) r-+ n[N (a) ] E ffi.

such that lim f(x) resp. lim f(x) is x=a x=a

lim n[N (a) ] resp. N(a)EN(a)

lim n[N (a) ] . N(a) EN

1. 7. 16 Exercise. If X is a topological space and f E ffi.x ,

<t> : X 3 x r-+ lim f(y) resp. ¢ : X 3 x r-+ lim f(y) y=x y=x

is lsc resp. usc. 1. 7. 17 Exercise. (Weierstra:B) If (X, d) is a compact metric space and {xn}nEN C X, for some x in X and some sequence {nk}kEN'

1. 7. 18 Exercise. If (X, d) is a compact metric space and U is an open cover of X, there is a positive rS such that for each x in X and some U in U, B(x, rSt c U.

Section 1 .7. Miscellaneous Exercises

[Hint: Lebesgue's Covering Lemma (1 .2.39) applies . ]

1. 7. 19 Exercise. The product topology for a Cartesian product

clef X X = I' Er Xl'

41

of topological spaces Xl' : a) is the weakest topology with respect to which all the projections 11").. : X 3 x �f {Xl' } f-t x).. E X).. are continuous; b) consists of the set of all unions of finite intersections of sets of the form 11"A 1 (U) , ). E r, U E 0 (X).. ) .

1.7.20 Exercise. If X is a set , Y has the topology T, and F e yX , the

set { n fs- 1 (Us ) : #(a) E N, fs E F, Us E T } is a topology for X and sEa

is the weakest topology with respect to which each f in F is continuous.

1. 7.21 Exercise. If (X, d) and (Y, D) are complete metric spaces and F e C(X, Y) , then F is normal iff":

a) for each x in X, { f(x) : f E F } is precompact in C(X, Y) ; b) F is equicontinuous at each x in X, i .e. , if E > 0 then for some positive

J and all f in F, {d (x, x') < J} ::::} {D [J(x) , ! (x') ] < E} . 1. 7.22 Exercise. If K is compact and f is usc resp. lsc on K, for some a in K, f(a) = sup f(x) resp. f(a) = inf f(x) . xEK xEK 1. 7.23 Exercise. (Tietze's Extension Theorem) If K is a compact subset of a locally compact space X, and f E C(K, JR.) , for some F in

Coo (X, JR.) �f { f : f E C (X, C) , supp (I) E K (X ) } ,

FiK= f. [Hint: The assumption f(K) C [-1 , 1] is justifiable. Applied to

+ clef { 1 } clef { 1 } K = x : f(x) 2': 3 resp. K- = x : f(x) :::; - 3 '

1.2.41 implies there is in Coo(X, C) an h such that

{ I if x E Kh (x) = -

31

3 if x E K+ 2 1 I f(x) - h(x) 1 :::; 3 ' on K Ih (x) 1 :::; 3 on x .


The argument above applied to f - II yields an h . Induction 00

yields a sequence {fn} nEW Finally, F � L fn exists and meets n=1

the requirements. ]

1 .7.24 Exercise. a) If A 3 ), r-+ f>. E usc(X) resp. A 3 ), r-+ f>. E Isc(X) is a net , and f>. ..l- f resp. f>. t f, then f E usc(X) resp. f E Isc(X) . b) If X is a locally compact Hausdorff space and 0 :::; f E lsc( X) there is a net

00 A 3 ), r-+ f>. E Coo (X, JR.) such that f>. t f. c) If f �f " x ( J ' f is lsc � n,n+ l n=-CX) but for no net A 3 ), r-+ f>. E Co(JR., JR.) is f>. t f valid.

[Hint: a) { x f (x) � a } = n { x : f>. (x) � a } . AEA

b) If S �f { <p <P E Coo (X, JR.) , . , f>. t f ·

1.7.25 Exercise. If X �f { (x, y) : x E JR., O :::; y E JR.} (the upper halfplane) , a topology T for X is defined by the following base B : B is in B iff either for some positive r, B �f { (x, y) : (x - a) 2 + (y - b)2 < r2 :::; b2 } or B = {(a, O)}U { (x, y) : (x - a)2 + (y - b)2 < b2 } . Thc space X is separable but not second countable.

1 . 7.26 Exercise. a) If 1 .2 . 19a)-d) are taken as the axioms for a closure operation and a set A is defined to be closed iff A = A, the set of complements of closed sets satisfies the axioms 1.2 .1 for open sets . b) (S . Mrowka) The axiom d) and the added axiom A u B = A u A u B imply a)-c ) . 1 .7.27 Exercise. A topological space (X, T) i s a Hausdorff space iff: a) each filter contained in !fj(X) converges to at most one point ; b) each net A r-+ X converges to at most one point ; c) the intersection of the set of all closed neighborhoods of a point x is x itself.

1 . 7.28 Exercise. If X is a locally compact space, y tJ. X, and

X* �f Xu{y}, then: a) { X* \ K : K E K(X) } u O(X) is a base of neighborhoods for a topology for X* ; b) if X is a Hausdorff space so is X* ; c) in the given topology, X* is compact (X* is the one-point compactification of X) .

1 . 7.29 Exercise. If X = JR. and X* is the one-point compactification of X, for no f in C (X*, JR.) is fl x = id : there is no continuous extension of sin : X 3 x r-+ sin x to X* , v. 3 .7.19.

Section 1. 7. Miscellaneous Exercises 43

1. 7.30 Exercise. If X is a complete metric space, F e C(X, q , and for each x in X and each f in F, sup If (x ) 1 < 00, then on some nonempty open

JET subset V of X , sup If (x) 1 < 00.

xEV fEF

[Hint: For some n in N, { x : sup If (x) 1 ::;; n } O "1 0.]

JET

1. 7.31 Exercise. If f E JR.lR , N 3 k > I, and for each y in JR., # [J-l(y)] = k,

f is not continuous . [Hint: The intermediate value property of continuous functions applies.]

1. 7.32 Exercise. a) If

S �f {xn}nEN C JR., L �f lim xn, l �f lim xn , E > 0,

then: n--+ CXJ n --+ CXJ

# ({ n : Xn > L - E }) = No , # ({ n Xn > L + E }) < No ,

# ({ n : Xn < l + E }) = No , # ({ n Xn < l - E } ) < No ,

S· c [l, L] . What are l, L, S· , sup (S· ) , and inf (S·) when, for n in N,

Xn �f (- I)nn, Xn �f n ((_ 1 )n - 1) , Xn �f n ( (-I )n + 1) and �f 1 ( -1 )'n _ 1 ( -1 ) m �f { Ym Ym - + -- , Zm - - + -- , Xn -m m Zm

if n = 2m + I ? if n = 2m .

b) If n : A r-+ JR. is a net , L �f lim n(>.) resp. l �f lim n(>.) , and E > 0: bl) AEA AEA

for each f.-l in A and some >. such that >. >- f.-l, n(>.) >- L - E; for some f.-l in A, if >. >- f.-l, then n(>.) 'I- n(f.-l ) ; b2) for each f.-l in A and some >. such that >. >- f.-l, n(>.) -< l + E; for some f.-l in A, if >. >- f.-l, then n(>.) -A n(f.-l) .

For a subset S of JR.,

N (S) �f { x : x E S n (0, 00) , for infinitely many n in N, nx E S } .

1. 7.33 Exercise. If S c [0, (0) and S is open and unbounded, then N (S) = [inf(S), (0 ) .

[Hint: For m in N, if Sm �f { x : mx E S } , Sm is open, U Sm 00 00

is dense in [inf(S) , (0) , and N (S) = n U Sm; 1.3.6 applies.] n= l Tn=n

2 Integration

2.1. Daniell-Lebesgue-Stone Integration

1 The function I I : JR. 3 x r-+ I x l is in C(JR., JR.) , whence inf(x, O) = "2 (x - Ix l ) 1 and sup(x, O) = "2 (x + I x l ) depend continuously on x. Therefore

L �f C( [a, b] , JR.)

is a function lattice , more particularly a vector space closed with respect to the operations inf (1\) and sup (v) : {f, g E L} ::::} {f 1\ g, f V g E L} . Furthermore, if fn ..l- 0 , then fn � 0 (cf. 1.2 .46 and 1.2.48) .

b b For the Riemann integral 1 ' the map 1 : L 3 f r-+ 1(1) �f 1 f(x) dx

is a nonnegative linear functional, i .e . ,

{f 2': O} ::::} {1(1) 2': O} (1 is nonnegative), 1(0'1 + (Jg) dx = od(l) + (J1(g) (1 is linear) .

(If f ::;; g then 1(1) ::;; 1 (g) : 1 i s a monotonely increasing functional . ) Furthermore, 1 .2.46 implies

{fn ..l- O} ::::} {1 (In) ..l- O} . (2 . 1 . 1 ) The purpose of this Chapter i s to develop a general theory of inte

gration based on the paradigm above, i .e . , a nonnegative linear functional 1 defined on a function lattice L and subject to the condition (2 . 1 . 1 ) .

-x For a set X, L denotes a function lattice in JR. , i .e . , a)

{f, g E L} ::::} {J 1\ g, f V g E L}

(L is a lattice) ; b) when f, g E L and a E JR., then O'f E L; c) for a in JR., the conventions

44

{ 0 if a = 0 ±oo ± a = a ± 00 = ±oo, and a · ±oo = ±oo if a > 0

=t=oo if a < 0

Section 2 . 1 . Daniell-Lebesgue-Stone Integration

are observed and when {f, g} C L

f + g : X 3 x r-+ {� (x) + g(x) iff f(x) + g(x) i- ±oo + (=Foo) otherwise.

45

Hence L is a vector space and f + (- f) is the additive identity of L: f + (- f) == o.

The following discussion takes place in the context of a set X , a func-x tion lattice L contained in ffi. , a nonnegative linear functional

I : L 3 f r-+ I (f) E ffi..

Thus : a) {f 2': O} ::::} {I(f) 2': O} ; b) when {a, b} C ffi. and {f, g} C L, then

I(af + bg) = aI(f) + bI(g ) .

The crucial added assumption about I is:

A functional I for which the preceding obtain is a Daniell-Lebesgue-Stone functional, abbreviated as DLS functional .

The next results lead to a function lattice L1 such that L C L1 C JRx and to an extension of I from L to a nonnegative linear functional

Various forms of abstract integration are special cases of the general context provided. The associated concepts and theories of measure and measurability (v. Section 2.2) are derivable as well.

The motivation for what follows is the improper Riemann integral. For example, when 0 < a and

[ 1 clef {x-a F : 0, 1 3 X r-+ F(x) =

00 if x > 0 if X = 0

the determination of the improper Riemann integral 1 1 F(x) dx (improper

because F is unbounded and JR-valued on [0, 1 ] ) is made by evaluating the

proper Riemann integrals 1 1 F(x) 1\ n dx �f 1 1

Fn (x) dx, n E N, and calculating the result as n t oo. The sequence {F n} nEN increases monotonely

which implies that lim 1 1 Fn (x) dx exists in JR. According as a < 1 or n--+CX) 0

a 2': 1 the limit is in ffi. or 00 (in JR) .

46 Chapter 2. Integration

Limits of monotonely increasing sequences of continuous (hence Riemann integrable) functions provide a larger class of functions to which an extension of the Riemann integral is definable. In the context of L , when L 3 In t I, the analogous procedure is to

define l(f) �f lim I (f n) [= sup I (f n)] . The set of functions that are the n--+CXJ n limits of monotonely increasing sequences drawn from L is denoted Lu .

Since I maps L onto JR., if L 3 In t i E Lu , then

-00 < I (fn) < 00 and - 00 < l(f) ::;; 00.

As 2 .1 .2 - 2 .1 .4 show, i f {In}nEN c Lu and a 2': 0, then In + fm E Lu , aln E Lu and, i f In t I , then I E Lu .

However, although F in the paradigm above belongs to the corresponding Lu, -F does not since no function continuous on [0, 1] lies below -F: Lu is not necessarily closed with respect to multiplication by nega-tive constants. (If X consists of a single element and L �f JR.x (= JR.) , then

-x Lu = JR. , which is closed with respect to multiplication by arbitrary real constants. )

Owing to the preceding observations, there is a temptation to extend L to the set of limits of all monotone sequences-not only monotonely increasing sequences. However, the goal of the DLS construction is achieved-and more economically-without resorting to the more elaborate procedure.

2.1 .2 THEOREM. IF {In}nEN , {gm}mEN C L, AND In t I, gm t I, THEN

PROOF . Since In ::;; I = lim gm , m--+=

Hence lim I (gm) 2': lim I (f n) . The argument is symmetrical in the pair Tn --+ CXJ n --+ ex)

{{I"}nEN , {gm }mEN } · D [ 2.1 .3 Note. Thus if

L 3 In t I and l(f) �f lim I (fn ) , n--+=


then 1(1) (in ffi:!) is independent of the sequence {fn}nEN . Hence 1 is a map from Lu to ffi:.]

47

2.1 .4 THEOREM. FOR f AND g IN Lu AND a IN ffi.+ : �) f + g E Lu , af E Lu , AND 1(1 + g) = 1(1) + 1(g) , 1(af) = afJ, I .e . , 1 IS ADDITIVE

. 1\ AND semzhomogeneous ; b) f g E Lu ; c) V

{f � g} '* {1(1) � 1(g) } ;

� � d) IF Lu 3 fn t f, THEN f E Lu AND 1 (In ) t 1(1) .

PROOF . a) In L there are sequences {fn}nEN and {gn }nEN such that

fn t f, gn t g, fn + gn t f + g, 1 (In ) t 1(1) , 1 (gn) t 1(g ) ,

1(1) + 1(g) = lim 1 (In) + lim 1 (gn) = lim 1 (In + gn) = 1(1 + g ) . n�= n�= n�=

Since afn t af and 1 (afn ) = a1 (In) , 1(af) = a1(1) . 1\ 1\

b) If L 3 in t f, L 3 gn t g, and f(x) .::;: g(x) , then L 3 fn v gn t f v g · c) If L 3 fn t f and L 3 gn t g, then

fn(x) 1\ gn (x)

< { fn(x) � fn+ 1 (X) = fn+ 1 (X) 1\ gn+ 1 (X) if fn+ 1 (X) .::;: gn+ 1 (X) , - gn (X) '::;: gn+ 1 (X) = gn+ 1 (X) 1\ fn+ 1 (:r) if gn+ 1 (X) .::;: fn+ 1 (X) ,

whence fn 1\ gn � fn+ 1 1\ gn+ 1 . If fn 1\ gn t k, then k .::;: f 1\ g. If

k(x) < f(x) 1\ g(x) ,

then for large n, k(x) < fn(x) 1\ gn(x) .::;: k(x) , a contradiction :

fn 1\ gn t f 1\ g .

Hence 1(1) = lim 1 (gn 1\ fn) '::;: lim 1 (gn) t 1(g) . n --+ CXJ n--+ ex) d) For n in N, L contains a sequence {gmn},.nEN such that

Hence, if kn �f sup { % : 1 .::;: i , j '::;: n }, then L 3 kn :::; kn+ 1 .::;: fn+ 1 :::; f . For some k in Lu, kn t k and k :::; f . If m .::;: n, then gmn .::;: kn , whence for


each m, fm = lim gmn :::; lim kn = k :::; I· Thus I = lim 1m :::; k :::; I n--+ CXJ n --+ ex) Tn --+ CXJ

and so I = k (E Lu ) . Furthermore, since kn t k = I,

� -- � � 1 (kn ) t 1(k ) = 1(f) , 1 (kn ) � 1 (fn ) � 1(f) ,

whence 1(fn ) t 1(f) . D The next development is the basis for the construction of L1 , the cen

tral aim of the discussion to this point .

-x 2.1 .5 DEFINITION. FOR I E ffi. , WHEN { U U E Lu , U 2': I } = 0 ,

1(f) �f 00 .

WHEN { U : U E Lu , U 2': I } 1- 0 , 11 �f inf { 1(U) U E Lu , U 2': I } . FURTHERMORE, l(f) = - [1( -f)] . 2.1 .6 THEOREM. FOR THE FUNCTIONAL 1:

a)

{I � g} '* { {1(f) � 1(g) } 1\ {l(f) � I(g) } } , {a 2': O} '* {1(af) = a1(f) } ,

i .e . , 1 AND I ARE NONNEGATIVE AND semihomogeneous ; b) 1(f + g) � 1(f) + 1(g) , resp. l(f + g) 2': HI) + l(g) , IF THE RI.9HT

MEMBERS OF THE PRECEDING INEQUALITIES ARE DEFINED, i .e . , 1, IS ESSENTIALLY SUB ADDITIVE resp. I IS ESSENTIALLY SUPERADDITIVE;

c) l(f) � 1(f) ; d) {I E Lu} '* {1(f) = l(f) = 1(f) } ;

e) { {O � In } 1\ {I �f �In } } '* {1(1) � �1 (fn) } ;

PROOF . a) If I � g and Lu 3 U 2': g , then U 2': I , whence I(f) � 1(g) . If a 2': 0, then {Lu 3 V 2': f} {} {Lu 3 aV 2': af} and 1(aV) = a1(V) . Hence

1(af) = a1(f) . b) If I � h E Lu , g � k E Lu , then I + g :::; h + k E Lu and

1(f + g) � 1(h + k) = 1(h) + 1(k) ,

Section 2 . 1 . Daniell-Lebesgue-Stone Integration 49

whence 1 (f + g) :s; 1 (f) + 1 (g) if the right member of the inequality is defined; the inequality l(f + g) 2: l(1) + l(g) follows from similar arguments . (When L �f C( [O, IJ , ffi.) , I i s the Riemann integral, and

f : [0, 1] 3 x r-+ { -x-1 �f x > 0 00 If x = 0

: [0 1] 3 x r-+ { x-2 if x > 0 g , ·f 0 ' 00 1 X =

then 1(f) = -00 and 1(g) = 00, whence 1(f) + 1(g) is not defined. ) c) If 1(f) = 00, the result is automatic. If 1(f) < 00, then

1(f) + 1(-I) is defined, whence

0 = 1(0) = 1[1 + (-f)] :s; 1(f) + 1( -I) = 1(f) - [-1( -I)] = 1(f) - l(f) ;

d) Since f :s; f, if f E Lu, then 1(f) :s; J(f) . If f :s; g E Lu, then

J(f) :s; J(g) ,

whence J(f) :s; 1(f) : 1(f) = J(f) . Owing to the linearity of I and the last equality, if g E L,

1( -g) = J( -g) = I( -g) = -I(g ) .

Thus l(g) = -1( -g) = - [-I(g)] = I(g) . I f f E Lu and L 3 gn t f, then 1 (gn) = I (gn) :s; J(f) and so

l(1) :s; 1(f) = J(f) = lim I (gn ) = lim 1 (gn ) :s; l(f) : n --+ ex) n--+ CXJ

1(f) = l(f) = J(f) . e) If any 1 (fn ) = 00, the implication is automatic. If each 1 (fn) is

finite and E > 0, then for some gn in Lu,

00 Moreover, by virtue of 2. 1 .4d) , g �f 2: gn E Lu and

n=1 00 00

J(g) = 2: J(gn ) :s; 2: 1 (fn) + E . n=1 n= 1


00 Furthermore, g 2': I and hence 1(1) :::; I(g) :::; L 1 (In ) + E. D

n= 1 [ 2.1 . 7 Note. Although the implication

{I 2': �In} * {1(1) 2': 1 (t, ln) }

is valid, the implication

is not .]

-x 2 .1 .8 DEFINITION. A FUNCTION I IN ffi. IS IN L1 IFF

-00 -00. Hence, if I E Lu and 1(1) < 00, then -00 < 1(1) = l(l) = 1(1) = 1(1) < 00, i .e . , if I E Lu and 1(1) < 00, then I E L1 . )

The properties listed in 2.1 .6 for 1 lead to

2.1.9 THEOREM. a) L C L1 AND J I L = I; b) L1 IS A VECTOR SPACE AND J IS LINEAR ON L1 ; c) I E L1 IFF FOR EVERY POSITIVE E AND SOME g AND -h IN Lu , h :::; I :::; g AND I[g + (-h) ] < E; d) L1 IS A FUNCTION LATTICE; e) THE MAP J IS A NONNEGATIVE (LINEAR) FUNCTIONAL ON L1 ; f) IF L1 3 In t I AND FOR ALL n, J (In) :::; M < 00, THEN l E U AND J (ln) t J(I) . CONVERSELY , IF I E L1 AND L1 3 In t I, THEN

J (In) t J (I) .

clef PROOF. a) If I E L and gn = I, n E N, then L 3 gn t I, whence I E Lu: L C Ln · Furthermore, 1(1) = 1(1) : Il L = I, whence

00 > 1(1) = 1(1) = l(l) > -00 : I E L1 .

A similar argument shows that J I L= I.


b) If f E Ll , C E JR., then { 1(cl) = infLu 3h?:f J(ch) = c infLu3h?:f J(h) = c1(1) l(cf) = -1(-cl) = - [-c1(1)] = c1(1) = cl(l)

whence cf E LI .

if c 2': 0 if c < 0 '

51

If {f, g} e LI , then the preceding argument and the subadditivity of 1 imply

1(1 + g) � 1(1) + 1(g) = J(I) + J(g ) , -l(f + g) = 1(-f - g) � -1(1) - 1(g) = -J(I) - J(g) ,

1(1 + g) 2': l(l + g) 2': J(I) + J(g) 2': 1(1 + g ) , whence J i s a linear functional on Ll .

c) If f E Ll and E > 0, then for some g in Lu , g 2': f and - E I(g) < J(I) + 2 .

- E Similarly for some -h in Lu , -f � -h and I(-h) < J(-f) + 2 . Hence h :::; f :::; g and

J[g + (-h)] = J(g) + J(-h) :::; J(I) + J(-I) + E = J(O) + E = E.

Conversely, if there are g and -h as described, then, as noted after the introduction of Lu , for any k in Lu, -00 < J(k) . Since

- - -I(g + (-h) = I(g) + I(-h) < E,

- -both I(g) < 00 and I( -h) < 00. Hence

1(1) � l(g) < 00, 0 :::; 1(1) - l(l) = 1(1) - [-1(-I)] ,

� J(g) + i(-h) = J[g + (-h)] < E .

d) If Lu 3 hi :::; Ii :::; gi E Lu , i = 1 , 2, then

When {h , h} e LI and gi and hi , i = 1 , 2, are chosen as in c) , then

- (hl � h2) = -hI � - h2 E Lu, J (g<g2) + J [- (hl � h2) ] :::; J(gl - hI ) + J(g2 - h2 ) < E,


1\ 1 whence h h E L . V e) Since I is nonnegative on L, I is nonnegative on Lu . Since J I Lu = I,

J is nonnegative on L1 . 00

f) If g �f I - h , then g � 0 and g = L (In+ 1 - In) . The subadditiv-n=1

ity of I, 2.1 .6d) , and the hypothesis J (In ) :s; M < 00 imply

00

l(g) :s; '" J (In+ 1 - In) = lim J (In ) - J (ld , � n--+CXJ n=1

1(1) = 1 (h + g) :s; J (ld + l(g) :S; lim J (In ) < 00, n ...... oo

HI) � I (In ) = J (In ) , I(I) � lim J (In ) � 1(1) ,

n ...... oo

whence I E L1 and J(I) = lim J (In ) . n ...... oo

If I E L1 , then, because J is nonnegative and 0 :s; I - In E L1 ,

O :S; J (I - In) = J(I) - J (ln) :

J (In ) :s; J(I) < 00. The argument in the previous paragraph applies.

[ 2 .1 .10 Note. The result 2 .1 .9f) is Lebesgue 's Monotone Convergence Theorem as it applies to the functional J.]

D

2 .1 .11 DEFINITION . THE SET Lui CONSISTS OF ALL g SUCH THAT FOR SOME SEQUENCE {gn}nEN CONTAINED IN Lu , gn ..l- g , -00 < lim I(gn) ,

n ...... oo AND 1(gd < 00. 2 . 1 . 12 Exercise. Lui C L1 .

[Hint: If g E Lui and gn are as described in 2 .1 .11 , for each n,

Lebesgue's Monotone Convergence Theorem applies.] Although any function considered in the preceding development is ffi:

valued, some of the functionals introduced are JR.-valued and others are ffi:-valued. In sum:

• Any function in L, Lu , L1 or Lui is ffi:-valued. • The functionals I and J are JR.-valued. • The functionals I and 1 are ffi:-valued.

Section 2 . 1 . Daniell-Lebesgue-Stone Integration 53

2.1 .13 THEOREM. AN I IS IN U IFF FOR SOME g IN Lui AND SOME NONNEGATIVE FUNCTION P IN Ll , J (p) = 0 AND 1 = g - p. PROOF . If I = g - p as described, then 2 .1 .9 shows I E Ll .

If I E Ll , then 1(1) E ffi. and for n in N, and some In in Lu , In 2': I and 1 (1n ) < 1(1) + .!. . Thus {gn �f h /\ . . . /\ In } is a sequence such n nEN - clef 1 that for some g in Lui , gn ..l- g. Moreover, I(g) = J(I) , 0 .:s: g - I = p E L , J(p) = 0, and I = g - p. D

2.1 .14 Exercise. a) (Fatou's Lemma) If 0 .:s: In E Ll , n E N, then

J ( lim In) .:s: lim J (In ) . n --+ CXJ n --+ CXJ

b) If I ln l .:s: I h l , n 2': 2, then J ( lim In) 2': lim J (ln ) . n --+ CXJ n --+ CXJ

The functional J is an emphasizer. It makes inferior limits more inferior and superior limits more superior.

[Hint: a) Only if lim J (In) < 00 is an argument needed. If n ...... = clef . I k 1').T clef h gk = mf n , E n , and gkn = h /\ . . . /\ h+n , t en n>k

2.1 .12 implies gk E LI , k E N, and gk t lim In . Furthermore, n ...... =

whence Lebesgue's Monotone Convergence Theorem [2 . 1 .9f)] applies. b) The mnemonic lim ( · ) = - lim (- · ) and 2 .1 .14a) apply.]

2. 1 .15 Exercise. (The Dominated Convergence Theorem) If

and lim In �f I, then I E Ll and lim J (In ) = J(I) . n--+CXJ n--+CXJ [Hint: The results in 2 .1 . 14 apply.]

Lebesgue's Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem are central features of the DLS functional J. Each result can be described crudely by the mnemonic: lim J = J lim.


In particular, Fatou's Lemma is an extension of the fundamental inequality { { {fn}nEN C L} 1\ {fn ..l- O} } ::::} {I (In ) ..l- O} with which the development began.

2.1.16 Exercise. When X � JR., L is the set of JR.-valued functions f such that supp (I) is compact and f is Riemann integrable on an interval [a, b] containing supp (I) , and I is the Riemann integral, if {fn } nEN C L and fn ..l- 0, then J fn (x) dx ..l- o .

[Hint: Since -fn t o, 2.1 .19f) applies .]

2.1 .17 Exercise. When X �f N, L �f Coo (N, JR.) , and I is defined by 00

I : L 3 f r-+ L f(n) , then: a) U is the set of (real) sequences {an }nEN n=1 00 such that L lan l < 00; b) the Dominated Convergence Theorem implies

n=1 00 that if L an is an absolutely convergent series and 11" is an autojection

n=1 00 00 (permutation) of N, then L a7r(n) = L an .

n=1 n= 1 2.1 .18 Exercise. If {fn }nEN C C( [O, 1] , JR.) , sup I fn (x) l :::; Mn , and xE [0, 1j

00 L Mn < 00, n=1

00 then f : [0, 1] 3 x r-+ L fn (x) is continuous (and in L1 ) . n=1 For a set E, the characteristic function of E is

x (x) �f {I if x E E . E 0 otherwise

2.1 .19 Exercise. a) If X �f JR. and L is the set of all real linear combinations of characteristic functions of bounded right-open intervals, then

N N

I : L 3 L anX[an , bn ) r-+ L an (bn - an) n=1 n=1

is a D LS functional . b) If X �f JR., L �f Coo (JR., JR.) , and I is the Riemann integral, then I is a DLS functional. c) The L1 created via a) is the same as the L1 created via b) .

Section 2.2. Measurability and Measure 55

2.1 .20 Exercise. a) If X �f JR.n and L is the set of all real linear combinations of characteristic functions of half-open n-dimensional intervals,

N N N

then I : L 3 � anx [ X�=l [an ,bn )] f-t � an 11 (bn - an ) is a DLS func-

tional . b) If X �f JR.n , L �f Coo (JR.n , JR.) , and I is the n-fold iterated Riemann integral, then I is a DLS functional . c) The L1 created via a) is the same as the V created via b) . 2.1 .21 Exercise. a) For an arbitrary set X, a in X, and the function sublattice L of JR.x , Ja : L 3 f f-t f( a ) , the Dirac functional at a, is a DLS functional. b) What are Lu and V ? 2.1.22 Exercise. a) For a set X in the discrete topology !fj(X) , Coo (X, JR.) is a function lattice Ld. b) The functional L : Ld 3 f f-t L f(x) is a DLS

xEX functional . c) What are (Ld)u and L�? (The vector lattice L� is usually denoted £1 (X) . )

2.2. Measurability and Measure

The works of Lebesgue and Caratheodory in the field of measurable functions, measurable sets, and measures (of measurable sets) are natural developments from the DLS functional . In the DLS development of these concepts, measurability of functions is introduced first . It takes a useful form in terms of the following

2.2.1 DEFINITION. WHEN {a, b, c} c JR.,

mid (a, b, c) �f (a 1\ b) V (a 1\ c) V (b 1\ c) �f mid (a, b, c) E ffi..

CORRESPONDINGLY , FOR FUNCTIONS f, g, h (IN JR.X ) AND x IN X ,

mid (I, g, h ) (x) �f mid [J (x) , g (x) , h (x) ] .

[ 2.2 .2 Note. Although mid (a , b , c ) E {a , b , c} , mid (I, g, h) i s not necessarily in {f, g, h} , e.g. , if

X = JR., f(x) = l x i , g(x) = I , h(x) = 3 - Ix l .]

2.2.3 Exercise. a) mid (a , b, c) = (a V b) 1\ (a V c) 1\ (b V c) ; for functions f, g, h, if g :s: h, then g :s: mid (I, g, h) :s: h. b) A function lattice is mid-closed. c) If L C jRx and L is closed with respect to addition and multiplication by scalars , then L is a function lattice if some nonzero constant


function is in L, each element of L is bounded in absolute value, and L is mid-closed, i .e . , {{f, g , h} c L} ::::} {mid (I, g , h) E L} .

[Hint: c ) I f I f I :::; M then If I = mid (M, f , -f) .]

2.2.4 Exercise. then: a)

-x For a set X, if a E ffi. \ {O} and {J, g, h , p, fn } C ffi. ,

{mid (af, g , h) = a mid (f' � , �) } , p + mid (I, g, h) = mid (p + f, p + g, p + h) ,

{h :::; h} ::::} {mid (h , g, h) ::;; mid (h , g, h) } ,

mid (h �h, g, h) = mid (h , g, h) � mid (h , g, h) ,

mid ( lim fn , g, h) = lim mid (In , g, h) . n --+ CXJ n --+ CXJ

b) If g(x) :::; h(x) , then

h 1\ f(x) - g 1\ f(x) { h(x) - g(x) = �(x) - g(x)

if mid (I, g , h) (x) = h(x) if mid (I, g, h) (x) = f(x) . otherwise

(2 .2 .5)

(2 .2 .6) (2 .2 .7)

(2.2.8)

(2.2.9)

(2 .2 . 10)

c) The functional mid is not additive: for some {f, g, h, p, q} contained in JRx , mid (p + q, g, h) i- mid (p, g, h) + mid (q, g, h) .

-x 2.2.11 DEFINITION. FOR A FUNCTION LATTICE L CONTAINED IN ffi. AND AN f IN ffi.X , f IS DLS measurable , i .e . , f E D(L) , IFF

{L 3 g :::; h E L} ::::} {mid (I, g , h) E L} .

A motivation for 2.2 .11 derives from consideration of the functions

and when 0 < a,

f : [0, 1] 3 x r-+ { I if x E � o otherwIse

d f { -a F : [0, 1] 3 x r-+ F(x) � � if x > 0

if X = 0

considered in the development leading to 2.1 .2 . Although f is bounded, it is not Riemann integrable (every upper

Riemann sum is I, every lower Riemann sum is 0) . If g(x) � 2x and h(x) � 3x, then g ::;; h , g and h are Riemann integrable, but mid (I, g , h)


is not Riemann integrable. Thus if L is the function lattice consisting of functions Riemann integrable on [0, 1] , f fails to belong to L not because f is unbounded but because the part of it that is trapped between g and h behaves badly with respect to Riemann integration: f is not Riemann measurable.

On the other hand, F is unbounded and yet , for the L where F is introduced, if L 3 g ::;; h E L, then {mid (F, g , h) , mid (-F, g, h) } c L so that ±F may be viewed as Riemann measurable .

In what follows, for the particular function lattice Ll , V (L 1 ) is denoted V.

[ 2.2.12 Note. Owing to 2.2.4, the spaces Lu, Lui , Ll , and V are mid-closed. In particular, Ll C V.]

2.2.13 THEOREM. a) THE SET V IS A FUNCTION LATTICE AND A monotone class of functions . b) {J E LI } {} {{f E V} 1\ { I f I E Ll } } . c)

{J E V} {} { { 0 ::;; g E Ll } '* {mid (I, -g, g) E Ll } } .

PROOF. a) By virtue of (2.2.8) and 2.1 .9d) , V is a lattice. If h , 12 E V and L l 3 g ::;; h E Ll , then for n in N,

cPin �f mid [Ii , -n( lg l + I h l ) , n( l g l + Ih l ) ] E Ll , i = 1 , 2, clef ·d ( h) 1 cPn = ml cPln + cP2n , g, E L ,

g ::;; cPn ::;; h, and cPn t mid (h + 12, g, h) .

Then 2 .1 .9f) implies mid (h + h, g, h) = lim cP" E LI : h + 12 E V. If n-+oo a i- 0, (2.2.5 ) implies ah E V: V is a function lattice.

If Ll 3 g ::;; h E Ll and V 3 fn t f, then mn �f mid (In , g , h) E LI and -00 < J(g) ::;; J (mn ) ::;; J(h) < 00. The conclusion follows from (2.2.9 ) . If V 3 fn ..l- f a similar proof applies. Thus V is, by abuse of language, lim- , lim- , and lim-closed.

b) The result follows from the formula: mid (I, - If I , I f I ) = f· c) '* : L l is a vector space. {=:: If Ll 3 P ::;; q E Ll , (2.2.6) implies

p + q . ( q - p q - p) . ( p + q ) -2- + mId f, - -

2-' -2- = mId f + -2- ,P, q .

p + q 1 Hence f + -2- E V. Since L e V, a) implies f E V. D


2.2.14 THEOREM. A NONNEGATIVE f IS IN V: a) IFF FOR ALL g IN L 1 , f /\ g E £1 ; b) IFF FOR ALL h IN L , f /\ h E L1 . PROOF. a) Since L1 C V, if 0 :s; f E V and g E L 1 , then g E V and 2.2 .13a) implies f /\ g E V. Since I f /\ g l :s; I g l E L1 , 2.2. 13b) implies f /\ g E L1 .

Conversely, if 0 :s; f and { g E L 1 } ::::} f /\ g E L 1 , the conditions

imply f /\ g E L 1 , f /\ h E L 1 , and g /\ h (= g) E L 1 , whence

(f /\ g) V (f /\ h) V (g /\ h) = mid (f, g, h) E L1 : f E V.

b) If 0 :s; f E V and h E L , then h E L 1 : a) implies f /\ h E L1 . Conversely, if 0 :s; f and for all h in L, f /\ h E L 1 , then, since L 1 is a

lattice, 2 .1 .9f) implies that if k E Lu and I(k) < 00, then f /\ k E L 1 and that if q E Lui, then f /\ q E L1 . If g E L 1 , then, according to 2 .1 . 13, for some q in Lui and some nonnegative r in L 1 , g = q - r and J(r) = o. Since g :s; q, as in (2.2 .10) ,

q /\ f (x ) - g /\ f(x) { q(x) - g (x) = r (x) 2': 0 = f (x) - g(x) 2': 0

o

if q(x) :s; f(x ) if g (x) :s; f(x ) :s; q(x) . (2.2. 15) if f (x) :s; g (x)

Owing to (2.2 .15) , 0 :s; f /\ q - f /\ g :s; r. Since J(r) = 0,

o :s; 1 (f /\ q - f /\ g) :S; 1(r) = J(r) = 0,

i.e. , f /\ q - f /\ g E L1 . However, f /\ q E L 1 and so f /\ g E L1 . Thus a) implies f E V. D

2.2.16 DEFINITION. A SUBSET 5 OF !fj(X) IS A ring resp. a-ring IFF 5 IS CLOSED WITH RESPECT TO THE FORMATION OF SET DIFFERENCES AND FINITE resp. COUNTABLE UNIONS, i.e. ,

{ {An }nEN C s } ::::} {AI \ A2 E S} , {N E N} ::::} {�

1 An E 5 } resp. 'v

N An E S.

A a-RING 5 IS A a-ALGEBRA IFF X E S. For any set X and a subset E of !fj(X) , aR(E) is, the a-ring generated

by E, i.e. , aR(E) is intersection of all a-rings containing E.


When X is a topological space, 5,(3(X) �f aR [O(X)] is the set of all Borel subsets of X.

When f E JR.x , a E JR., and 0 is one of <, :S;, >, 2:, =, 1-,

Eo (f, a) �f { x : x E X, f (x) 0 a }. (Hence Eo/- (f, 0) = supp (f) . ) When 5 i s a a-ring in !fj(X) an f in JR.x is by definition 5-measurable

iff r1 (5,(3 ) n Eo/- (f, 0) c 5. (The presence of Eo/- (f, 0) in the formulation is related to the possibility that X tJ- 5, i.e. , that 5 is not a a-algebra. )

When f E yX, T is a a-ring in !fj(Y) and Eo/- (f, 0) is meaningless, f is (5, T) -measurable iff f- l (T) C 5. In particular, when X is a topological space and 5 = aR[O(X)] [= 5,(3 (X) ] , an f that is (5, T )-measurable is Borel measurable .

[ 2.2 .17 Note. It is convenient to observe the convention whereby sans-serif fonts , e.g. , 5, denote subsets of !fj(X) and cursive fonts ,

x -x e.g . , S, denote subsets of JR. or JR. . For example, when 5 is a a-ring (cf. 2.2.16) contained in !fj(X), the set of functions that are in JR.x or JRx and are 5-measurable is denoted S.]

[ 2.2.18 Remark. Since X E F(X), 5,(3 (X) = aR [F(X)] .]

2.2.19 Exercise. In !fj (JR.n ) , 5,(3 (JR.n ) is the a-ring generated by: a) the set of all open balls; b) the set of all closed balls; c) the set of all half-open intervals; d) the set of all cubes .

A set E is a DLS measurable subset of X iff X E E D. The set of DLS measurable subsets of X is denoted D.

2.2.20 THEOREM. THE SET D IS A a-RING AND IF 1 E D, D IS A aALGEBRA.

PROOF. For C �f U An, from the equations nEN

X(AUB) = XA V XB, X(A\B) = X(AUB) - XB, XC = V X(An ) ' nEN and 2.2.13 it follows that D is a a-ring. When 1 E D, the formula

X(X\A) = 1 - XA

implies D is a a-algebra. D


2.2.21 THEOREM. IF f E JR.X , THEN f- 1 [S,(3 (JR.)] C D IFF FOR EACH REAL a, Edf, O') E D. PROOF. If f- 1 [S,(3(JR.)] c D, for each real a,

Edf, O' ) = r1 (-00, 0') E D.

Conversely, if each Edf, a ) i s in D and if a < b, then

f-1 ( [a, b)) = Edf, b) \ Edf, a) E D.

Since S,(3 (JR.) = aR ({ [a, b) : a < b, a, b E JR. } ) , r1 [S,(3 (JR.)] c D . D

2.2.22 Exercise. The assertion in 2.2.21 remains valid if E< is replaced by E<:: , E> , E� , or Eo/- (v. the last paragraph of 2.3.5 for a discussion of E=) .

2.2.23 THEOREM. IF 1 E D, THEN f E D IFF {a E JR.} ::::} {E< (f, O') E D} . PROOF. If f E D, then, since 1 E D, it follows from 2 .1 .9b) and 2 .1 .9d) that for n in N,

n(O' - f) E D, l /\ n(O' - f) E D, and [I /\ n(O' - f)] v O E D. Furthermore, X ( = lim [1 /\ n(O' - f)] V 0, whence 2.1 .9f) implies E< j,o:) n--+=

Edf, O') E D.

Conversely, if f E JR.x , then f = f V 0 + f /\ 0 �f f+ + f- , whence f is the difference of the two nonnegative functions: f = f+ - (-f- ) . For k and n in N, if r::k is the characteristic function of

n2n " k - 1 clef then � � r::k = r:: t f+ · For every a,

k=O

if 0' < 0 if a 2': 0 .

Hence, if E< (f, a) E D for every a, then f+ E D. A similar argument applies to -j- , hence to f- : f = f+ + f- E D. D

2.2.24 Exercise. If I f I :s; M < 00, then f;t: � f+ and f;; � f- · 2.2.25 Exercise. If 5 is a a-ring in !fj(X) and f E JR.x , then f E S iff for each a in JR., Edf, a) n Eo/- (f, 0) E 5: f is S-measurable iff for each a in JR., Edf, a) n Eo/- (f, 0) E S.

Section 2.2. Measurability and Measure

On D there is defined the JR.-valued set function:

if XE E L1 otherwise

61

Thus ,AD) c [0, 00] . Furthermore, if {En}nEN is a sequence of pairwise N

disjoint subsets in D and E �f U En , then E E D and " X t X . nEN � En E n=l Lebesgue's Monotone Convergence Theorem (2 .1 .9f) ) implies

IL (U nENEn) = f IL (En ) , n=l i.e., IL is a nonnegative countably additive set junction , that is to say, a measure. The triple (X, D, IL) is a measure space , i .e. , a set X, a a-ring or a a-algebra D contained in !fj(X), and a measure IL : D r-+ [0, 00] . 2.2.26 Exercise. a) For a measure space (X, D, IL) , 1L(0) = 0 and if D 3 A C B E D, then IL(A) :::; IL (B) . b) If

{En}nEN C D, En ::) En+1 ' n En = 0, and IL (E1 ) < 00, nEN

then IL (En) ..l- o. 2.2.27 Exercise. If 5 is a a-ring and ¢ : 5 r-+ [0, (0) is a finitely additive set junction, then ¢ is count ably additive iff

[Hint: For n in N,

E1 = (E1 \ E2) l:J . . . l:J (En- 1 \ En) l:JEn = U (En \ En+d , nEN n

¢ (Ed - L¢ (Ek \ Ek+d = ¢ (En ) .] k= l

For a measure space (X, 5, IL) , a set A in 5 is a null set iff IL(A) = o. The set N �f { N : N C A E 5, IL(A) � O } consisting of subsets of null sets in 5, gives rise to the completion 5, consisting of all sets of the form (E \ Nd u (N2 ) , E E 5, N1 , N2 E N . For F �f (E \ Nd u (N2 ) in 5,

ji(F) �f IL(E).


2.2.28 Exercise. If 5 is a a-ring resp. a-algebra, then 5 is a a-ring resp. a-algebra and (X, 5, Ii) is a measure space. (When 5 = 5, 5 and (X, 5, f.-l)

are complete . ) For two functions f, g in Ll , f � g iff f - g differs from zero only on

a null set . More generally, the notation a.e. is used to indicate that a statement is true almost everywhere, i.e. , except (at most) on a null set.

For example, lim fn = f a.e. or fn �. f means lim fn exists a.e. n4� n4� and lim fn � f. A function f in Ll is a null function iff J( lf l ) = O. n---+= 2.2.29 Exercise. a) The relation � is an equivalence relation. b) If f is a null function, g E ffi.x , and I g l :s; f, then g E Ll and g is a null function. c) An f is a null function iff f == O. d) The a-ring D is complete; e) D is a a-algebra if {f E L} ::::} {1 /\ f E L} .

[Hint: For b) , 0 :s; I( lg l ) :s; I( lg l ) :s; J(f) = 0 applies . For c) , if f :j:: 0, one may assume f V O :j:: O. Then for n in N,

E �f { n - X

applies. For d) , b) applies.]

: f V O > � } The quotient set L 1 / � of �-equivalence classes is usually identified

with Ll itself. (When the identification is not made, L 1 / � is sometimes denoted £1 . In all the results that are of interest in the remainder of this book, the distinction between Ll and £1 is of no consequence. Thus , for simplicity and ease of presentation, the distinction is henceforth ignored: Ll = £1 . )

2.2.30 Exercise. The map

d : Ll x Ll :1 (f, g) r--+ d(f, g) (� J( l f - g l ) �f I l f - g i l l is a metric for Ll . 2.2.31 Exercise. In the context of the DLS construction, L is dense in Ll metrized by d as described in 2.2.30.

[Hint: The criterion in 2 .1 .9c) for membership in Ll applies .]

2.2.32 THEOREM. METRIZED BY d AS IN 2.2 .30, Ll IS A COMPLETE METRIC SPACE.

PROOF. For a Cauchy sequence {fn}nEN in Ll , there is a sequence


such that if k 2': 2, then I l fnk - fnk- , l l l < 2-k . Lebesgue's Monotone Convergence Theorem (2. 1 .9f) ) implies that for some g in Ll ,

K

I fn, I + L Ifnk - fnk_ ' I t g. k=2

Thus for some f, lim [fn, + t (Ink - fnk_ , )] = lim fnK = f: f E D K4= K4=

k=2 and since I fnK I ::;; Ig l , the Dominated Convergence Theorem (2.1 . 15) im-plies f E Ll and lim I l fnK - f i l l = o. Consequently, for any n in N,

K4=

Since {f n} nEN is a Cauchy sequence in Ll , for large n and large K, I + I I is small, i.e. , lim I l fn - f i l l = o. D n4=

For a measure space (X, A, p,) , when {En} l<n<N is a set of pair-wise disjoint elements of A and {an L � n� N C JR., the linear combination N N L anx En �f s is a simple function . When L anp, (En) is meaningful, n= l n= l N Ix s (x) dp,(x) �f � anp, (En) [= J(s) ] . If 0 ::;; f E D,

Ix f(x) dp,(x) �f sup { Ix s(x) dp,(x) : s simple and 0 ::;; s ::;; f } . For f in D, f = f V 0 + f 1\ 0 and, when the right member below is meaningful, i.e. , when at least one of the terms in the right member is finite,

Ix f(x) dp,(x) �f Ix (I V O) (x) dp,(x) - Ix -(1 1\ O) (x) dp,(x) .

An f in D is integrable iff -00 < Ix f(x) dp,(x) < 00. The set o f integrable functions is denoted U(X, p,) . For f in Ll (X, p,) and E in 5,

[ 2.2.33 Note. The distinction between Ll (X, p,) and ,Cl (X, p,), the set of equivalence classes with respect to the relation � defined according to {f � g} {} {Ix I f - g l dp, = O} , is usually ignored.


The notation L1 (X, p,) emphasizes the involvement of the measure p" whereas the notation L1 used in the DLS construction suggests that there is no a priori measure used to define L1 . The next results sort out the relations between L1 (X, p,) and L1 .]

2.2.34 THEOREM. (Stone) IF p, IS THE MEASURE DEFINED ON D VIA THE FUNCTIONAL J AND L 1\ 1 C L, AN f IN D IS IN L1 (X, p,) IFF f E L1 , IN WHICH CASE Ix f(x) dp,(x) = J(f) .

PROOF. According to 2.2 .13b) , {J E L1 } {} { (f E D) 1\ ( I f I E L1 ) } . Since f = f+ + f- and I f I = f+ - f- , the discussion may be confined

to the set of nonnegative functions . If 0 :s; f E L 1 , then, since L 1\ 1 C L, as in the PROOF of 2.2.23, there

is a sequence {f n} nEN of nonnegative simple functions such that f n t f· Lebesgue's Monotone Convergence Theorem implies Ix f(x) dp,(x) � Ix fn(x) dp,(x) = J (fn ) t J(f) .

If Ix f(x) dp,(x) > J(f) , for some simple function s , O :s; s :s; f and J(s) > J(f),

in denial of the nonnegative character of J: Ix f(x) dp,(x) = J(f) . If o :s; f E L1 (X, p,) , for some simple functions { sn}nEN ' Sn t f and Ix sn (x) dp,(x) = J (s,, ) t Ix f(x) dp,(x) < 00 . Consequently, 2 .1 .9f) im-

plies lim Sn �f s E L1 and J(s) = r f(x) dp,(x) . On the other hand, the n---+= Jx preceding paragraph implies Ix s(x) dp,(x) = J(s) and so

Ix f(x) dp,(x) = Ix s(x) dp,(x) . I f f - s is not 0 a.e. , there emerges the contradiction

0 < r [J(x) - s(x)] dp,(x) :S; r [J(x) - s(x)] dp,(x) D JE> (f-S,O) Jx

2.2.35 THEOREM. IF (X, S, �) IS A MEASURE SPACE AND L IS THE SET OF SIMPLE FUNCTIONS s SUCH THAT Ix s d� E JR., THEN

I : L 3 s r-+ I(s) �f Ix S d� E JR.


IS A DLS functional, i.e. , A NONNEGATIVE LINEAR FUNCTIONAL SUCH THAT { sn ..l- o} ::::} {I (Sn) ..l- o} .

def { } n PROOF. If En = X : Sn (X) > 0 , then En ::) En+ 1 , and En = 0. Fur-nEN

thermore, � (Ed < 00, whence 2.2 .27 implies � (En) ..l- O. Since

and sup sn(x) ::;; sup S 1 (X) < 00, I (sn ) ..l- o. xEEn xEX D

In the context of 2.2.35, the DLS functional I gives rise to I, 1, J, D,

D, p" and, via 2.2.34, J . dp,. The next discussion deals with the relations between 5 and D resp. � and p, resp. L1 (X, O and L1 (which, according to 2.2.34 is L1 (X, p,) ) .

I f E E 5, then XE E D, whence 5 c D. I f 0 ::;; f E L1 (X, �) , there i s a sequence { sn} nEN of simple functions such that

0 ::;; Sn t f and Ix Sn d� t Ix f d�.

On the other hand, Lebesgue's Monotone Convergence Theorem for the DLS functional J derived from I and the associated measure p, implies Ix f d� = J(f) = Ix f dp, : L1 (X, �) C L1 . Since D is always complete, if 5 is not complete, 5 �D. However, since L is dense in both L1 (X, �) (metrized according to

and L1 , it follows that modulo null functions, L1 (X, �) = L1 = L1 (X, p,) . The previous developments offer justification for largely ignoring, in the

remainder of the book, the distinctions among U(X, �) , L 1 , and L1 (X, p,) . They also establish that Daniell's approach and Lebesgue's approach to integration (v. 2 .2 .50) are logically equivalent . As circumstances dictate, for exploring a problem one approach might appear superior to the other.

For example, as formulated by Kolmogoroff [Kol] , probability theory lends itself to Lebesgue's ideas (v. 2.2.50) . On the other hand, the elegant presentations of integration on locally compact groups [Loo, Nai, We2] and the Riesz Representation Theorem (2.3.2 below) are more readily understood from the perspective of Daniell's theory.

[ 2.2.36 Note. The basic results , Lebesgue's Monotone Convergence Theorem (2. 1 .9f) ) , Fatou's Lemma (2 .1 .14a) ) , and the


Dominated Convergence Theorem (2.1 .15) are valid for the functional J and hence for the functional Ix .]

2.2.37 Exercise. If ° ::;; f E D: a) for the map

¢ : D 3 E r-+ ¢(E) �f L f(x) dp,(x) ,

(X, D, ¢) is a measure space; b) if ° ::;; g E D, then Ix g d¢ = Ix gf dp" a relation that may be summarized by the equation d¢ = f dp" v. Section 5.8.

[Hint: For a) , Lebesgue's Monotone Convergence Theorem applies . For b) , the general result follows from the treatment of the special case when g is the characteristic function of a set E in D: g = XE·]

2.2.38 Exercise. If X � [0, 1] and, for f in C( [O, 1] , JR.) , I (I) is the

Riemann integral 1 1 f(x) dx of f, then D ::) aR[K ( [0, 1] ) ] = 5,6 ( [0, 1] ) , the

a-algebra of Borel sets in [0, 1] . If ° ::;; a < b ::;; 1, p,( [ a, b] ) = b - a. [ 2.2.39 Note. Although D is complete (v. 2.2 .28) , the following discussion concludes that D �5,6( [0, 1] ) and that 5,6 ( [0, 1] ) is not complete.]

2 .2.40 Example. When a E [0, 1 ) , the Cantor set Co: is constructed by deleting from [0, 1] a sequence of open intervals:

1 • 10 20 centered at - ; , 2 • Il l , 11 ,21 centered at the midpoints of the two components of

• • In1 , . . . , In,2n centered at the midpoints of the 2n components of

[0, 1 1 \ ['Q (�l; " ) 1 ; etc.


The intervals in any group In 1 , . . . , In,2n are of equal length, say an , 00

an > an+l , and 1 - a = L 2nan . It follows that Co: is a closed set and n=O

f.-l (Co: ) = a. If an == 3-n, then a = ° and there emerges the Cantor set

Co. A direct calculation shows that Co = {� �: : En = ° or 2 } . The

00 En 00 1 En • map Co 3 L -:;;: r-+ L "2 2n+ 1 carnes Co onto [0, 1] (v. 1 .2 .31) . Hence n= l n=l #( [0, 1] ) 2': # (Co) 2': #( [0, 1] ) : # (Co ) = c.

Because f.-l (Co) = ° and D is complete, !fj (Co ) c D. Furthermore,

2' :::; #(D) :::; # [!fj(ffi.)] = 2' .

Thus #(D) = 2' . On the other hand, as the next lines show, # [5,(3 (ffi.)] = c. Hence some

subset of Co is not in 5,(3 (ffi.) : 5,(3 (ffi.) is not complete and, to boot, D �5,(3 (ffi.) . Since every singleton set is in 5,(3 ( [0, 1] ) , # (5,(3 ) ( [0, 1] ) 2': c. Further

more, 5,(3 ( [0, 1] ) = aR ({ (a, b) : O :::; a :::; b :::; l , a, b E Q } ) �f aR (Eo ) , i.e. , 5,(3( [0, 1] ) is generated by the countable set Eo . For every ordinal number a in (0, Q), if Eo: is the set of all unions of count ably many set differences drawn from U E", then 5,(3 ( [0, 1] ) = U Eo: . However,

,,<0: o:<n

whence # [5,(3 ( [0, 1] )] = c.

# ( { a a < Q } ) :::; c,

Customarily, the complete a-algebra D in the context of 2.2.38 is denoted 5A ( [0, 1] ) , the set of Lebesgue measurable subsets of [0, 1] . By the same token, the measure f.-l is usually denoted A, and the measure space is ( [0, 1] ' SA ( [0, 1] ) , A) .

2.2.41 Example. For a in [0, 1 ) there is the Cantor function cPo: defined { 2k - 1 as follows : cPo: (x) = � if x E h,2n , 1 :::; k :::; 2n- 1 .

Thus far cPo: is defined only on [0, 1] \ Co: . However, if y E Co: , then

lim cPo: (x) �f cPo: (y) x -+ y xE [O , l ] \Ca

exists. The function cPo: is continuous , monotonely increasing, and


2.2.42 Exercise. a) <Pa ( [O , 1] ) = [0, 1] ; b) there is a measure space

( [0, 1] ' SA ( [0, 1] ) , f.L)

such that for [a, b) contained in [0, 1] , f.L ( [a, b) ) �f ). {<pa [[a, b)] } ; c) if

then '1jJa : [0, 1] r-+ [0, 1] is a homeomorphism.

2.2.43 DEFINITION. FOR A SET X, AN outer measure

f.L* : �(X) 3 E r-+ f.L* (E) E [0, 00]

IS A SET FUNCTION THAT IS countably subadditive , i .e. ,

MONOTONELY INCREASING, i.e. , {A C B} ::::} {f.L* (A) :::; f.L* (B) } , AND SUCH THAT f.L* (0) = O.

A SET H CONTAINED IN �(X) IS hereditary IFF

{ { E E H } !\ {F c En ::::} {F E H } .

WHEN A c �(X) , H (A) , THE hereditary a-ring of A, IS THE INTERSECTION OF ALL HEREDITARY a-RINGS CONTAINING A.

FOR A MEASURE SPACE (X, 5, f.L) ,

f.L* : H (S) 3 E r-+ inf { f.L(F) E c F E 5 }

IS THE outer measure induced by f.L on H (S) ;

f.L* : H (S) 3 E r-+ SUp { f.L(F) : E ::) F E S }

IS THE inner measure induced by f.L on H (S) . FOR A IN H (S) , A C IN 5 IS A measurable cover OF A IFF C ::) A AND

{5 3 B e e \ A} ::::} {f.L(B) = O}; A K IN 5 IS A measurable kernel OF A IFF K c A AND {5 3 B c A \ K} ::::} {f.L(B) = O}.

2.2.44 Exercise. a) In the context of the DLS construction,


is an outer measure. b) For a measure space (X, 5, p,) : bl ) if C1 and C2 are measurable covers of an A in H (5) ,

b2) i f A E H (5) , some K in 5 i s a measurable kernel of A; b3) i f A E H (5) and A is contained in a countable union of sets of finite measure (A is p,* -a-finite) , some C in 5 is a measurable cover of A. 2.2.45 Example. In the context of 2 . 1 .22, the measure corresponding to the functional L on the a-ring D is counting measure :

v : 5 "3 E t---+ {#(E) if #(E) E N . 00 otherwise

Contained in D is the sub-a-ring 5 consisting of (/) and the set of all finite or countable subsets of X. If #(X) > No , A C X and #(A) > No , no C is a measurable cover of A.

For a set X and an outer measure p,* , a subset E of X is Caratheodory measurable iff for all A in !fj(X)

p,* (A) = p,* (A n E) + p,* (A \ E) (2.2 .46)

(E splits every set A p,* -additively. ) The set of Caratheodory measurable sets is denoted C. (As 2.2.48 reveals, the condition (2.2.46) implies that p,* I c is count ably additive. ) 2.2 .47 Exercise. A set E is in C iff

{{P c E} !\ {Q C X \ E}} '* {p,* (Pl:!Q) = p,* (P) + p,*(Q) } .

2.2.48 THEOREM. THE SET C IS a-ALGEBRA, (X, C, p,* ) IS A MEASURE SPACE, AND C IS COMPLETE.

When E in (2 .2 .46) is E1 \ E2 , there emerges

p,* (P) + p,*(Q) = p,*(P) + p,* (Q n E2 ) + p,* (Q \ E2 ) = p,* [Pl:! (Q \ E2 )] + p,* (Q n E2 ) = p,* (Pl:!Q. )


Since El U E2 = Ell:.! (E2 \ El ) , the previous conclusion implies that to show El U E2 E C, it suffices to assume El n E2 = 0. Then for any set A, since

f.-l* [(A n Ed l:.! (A n E2)] = f.-l* { (A n Ed l:.! [ (A n E2) n Ed } + f.-l* { [ (A n Ed l:.! (A n E2)] \ Ed

= f.-l* (A n Ed + f.-l* (A n E2) , it follows that

f.-l* [A n (Ell:.!E2 )] + f.-l* [A \ (Ell:.!E2 )] = f.-l* [(A n Ed l:.! (A n E2)] + f.-l* [(A \ Ed \ E2] = f.-l* (A n Ed + f.-l* (A n E2) + f.-l* [(A \ E1 ) \ E2] ' (* )

On the other hand, f.-l* (A \ E1 ) = f.-l* [(A \ Ed n E2] + f.-l* [(A \ Ed \ E2]

= f.-l* (A n E2) + f.-l* [(A \ Ed \ E2] , whence the last two summands in the right member of (* ) may be replaced by f.-l* (A \ Ed. There emerges

f.-l* [A n (Ell:.!E2 )] + f.-l* [A \ (Ell:.!E2 )] = f.-l* (A n Ed + f.-l* (A \ Ed = f.-l* (A) :

Ell:.!E2 E C. Since C is closed with respect to the formation of set differences, in

duction shows that for N in N,

SN �f U En = Ell:.!U (En \ En-d �f U Fn E C, l�n�N 2�n�N l�n�N

Fn E aR (C) .

For N in N and A in !fj(X) , since each Fn is in C, induction and the monotone character of f.-l* imply that

f.-l*(A) = f.-l* (A n SN ) + f.-l* (A \ SN) , f.-l* (A n SN) = f.-l* (A n SN n FN) + f.-l* ( (A n SN) \ FN)

= f.-l* (A n FN) + f.-l* (A n SN-d N-l N

= f.-l* (A n FN) + L f.-l* (A n Fn ) = L f.-l* (A n Fn ) , n= l n= l

N f.-l*(A) = L f.-l* (A n Fn ) + f.-l* (A \ SN) ,

n= l 00

n=l


Hence Soo E C; if A = Soo the preceding calculations imply

i.e. , (X, C, p,* ) is a measure space. The definition of Caratheodory measurability implies: a) if E E C, then X \ E E C; b) if p,* (A) = 0, then A E C. The monotone character of p,* implies that C is complete. D

2.2 .49 THEOREM. IF 1 E D, AN E IN !fj(X) IS DLS MEASURABLE IFF E IS CARATHEODORY MEASURABLE WITH RESPECT TO THE OUTER MEASURE p,* OF 2.2.44. PROOF. If D is DLS measurable and p,* (A) = 00 the subadditivity of 1 implies p,* (A) = p,* (A n E) + p,* (A \ E) .

I f p,* (A) < 00 and E > 0, for some g in Lu , g 2: XA and

1 (XA) + E 2: I(g) ;::::: I(g /\ 1) = I (g /\ XD + g /\ X(X\D) ) 2: I (g /\ xD) + I (g /\ X(X\D)) 2: I (X(AnD) ) + I (X(A\D)) ,

i.e. ,

DLS measurability implies Caratheodory measurability. When C is Caratheodory measurable its DLS measurability is estab

lished by showing that Xc E D, i.e . , that for all g in L, Xc /\ g E L1 , (v. 2.2. 14a) ) .

Case 1 : I (xc) < 00. If E > 0 , for some h in Lu,

h 2: Xc and I( h) < I (Xc) + E < 00.

Since 1 E D, L1 '3 1 /\ g 2: Xc' Each of a)-e) below follows from its predecessors or from the statements made thus far:

a) 1 /\ h E D; b) B � E:2 (1 /\ h, 1 ) E D; c) h 2: XB E D; d) I (XB) :::; I(h) < 00, whence XB E L1 ; e) XB 2: Xc'


Since X E L1 , B is DLS measurable and hence Caratheodory measurable. The foflowing equations and inequalities are validated, i .a. , because:

• both B and C are Caratheodory measurable; • X E LI . B ' • I is subadditive; • for all sets A, I(X) = /-l* (A) (by definition):

/-l* (B) = /-l* (B n C) + /-l*(B \ C) ,

I (XB) = I (XB 1\ xc) + I (X(B\C)) = I (xc) + I (XB - xc) , I (xB - xc) S; I (xB) + I (-xc) = l (xB) - 1 (-xc) ,

= 1 (XB - xc) ·

Thus XB - Xc E L1, Xc = XB - (XB - Xc) E L1 , i .e . , C E D.

Case 2 : I (Xc) = 00 . Since the range of Xc is the set {O , I} , i t suf

fices to show: { {g E L} 1\ {O S; g S; I } } ::::} {Xc 1\ g E L1 } . If A = E> (g, O ) , owing to 2 .2 .22 , A E D .

Case 2a: X A E L 1 . Then A is DLS measurable, hence Caratheodory measurable and

Thus C n A is Caratheodory measurable and since

0 < X < X E L1 - (CnA) - A '

I (x ) < 00 The conclusion in Case 1 shows that (CnA) .

whence Xc 1\ g E L1 . Case 2b: XA E (D \ L1 ) . For E > 0, if A. � E> (g, E) , then

Hence X E L1 . The conclusion in Case 2a implies that C n A E D. Set-A, ting E at .!, n E N, yields the result . D n

Section 2.2. Measurability and Measure

[ 2.2.50 Note. The theory of integration, due to Lebesgue, per-12K mits meaning to be ascribed to 0 f(x) dx when

00 f (x) �f L an cos nx + bn sin nx,

n=l

00 and L I an cos nx + bn sin nx I < 00 for all x, even, as well might

n=l be the case, when f fails to be Riemann integrable. Such a func-tion is integrable when the symbol 12K f(x) dx is viewed as the Lebesgue integral.

Lebesgue's idea for a partition of the domain of a function f in ffi.[a,b] was contrapuntal to Riemann's. A Riemann partition of [ b] ' d . d b ' clef clef b d a, IS etermlne y pOlnts a = Co < Cl < . . . < Cn = an con-sists of the sets

a partition made without regard to the function f. Lebesgue partitioned the domain [a, b] of f by direct reference to f via its range. If p > q > r, the sets

Epq �f Edf, p) \ K; (f, q) , Eqr �f Edf, q ) \ Edf, r)

are disjoint . If f is sufficiently restricted, the sets Epq are in the a-ring S,B ' Whereas the length of Epq is not necessarily meaningful without modification of the notion of length, Lebesgue developed a theory that consistently assigns a value to the size of any Epq in S,B and indeed to any set E in S,B ' Successive elaborations by Lebesgue, Daniell, Stone, and others led to the development presented above and to the following discussion.]

2.2.51 DEFINITION. FOR AN ELEMENT E OF !fj(X) , A PARTITION

73


OF E IS A SET OF PAIRWISE DiSJOINT SETS SUCH THAT U P-y = E. IN -yEr THE CONTEXT OF A MEASURE SPACE (X, S , p,) , WHEN E E S , THE PARTI-TION P IS MEASURABLE IFF EACH P-y E S . THE PARTITION P IS FINITE IFF #(r) E N.

2 .2.52 Exercise. For the measure space (X, S, p,) , if 1 is measurable, o � f E L1 (X, p,) , P � {Pkh�k�n is a finite measurable partition of X, n and sp(f) � L inf f(x)p, (Pk) , then XEPk k=1

Ix f dp, = sup { sp : P a finite measurable partition of X } .

2.2.53 Exercise. a) Each fn in 1 .7.6 is Riemann integrable; b) Un}nEN is a bounded sequence; c) lim fn �f f exists; d) by contrast with 2 . 1.9 n---+= and 2 .1 . 15, f is not Riemann integrable; e) f is Borel measurable and for the measure space ( [0, 1] ' S,B, ),) , 11 f(x) d),(x) exists and

lim 11 fn (x) dx = 11 f(x) d),(x) = O. n---+= 0 0

2.2.54 Exercise. For f as in 2.2.53, in C ( [0, 1] , JR.) there is a sequence {gn }nEN such that: a) nl�� gn llQ!n[o, 1 j

exists and is f llQ!n[O, 1 j; b)

each gn is nonnegative; c) sup gn (x) = 1 ; d) 11 gn (X) dx > 0, n E N; e) O�x� 1 0 lim 11 gn (X) dx = 11 f(x) d),(x) , cf. 1 .3.5. n---+= 0 0

[ 2.2 .55 Note. As shown next , the results in 2.2.53 and 2.2.54 show that the DLS construction can provide a proper extension of the original DLS functional.

When the Riemann integral 11 . dx is used to define the (DLS )

functional I : C( [O, 1] , JR.) '3 f r-+ 11 f(x) dx E JR., the DLS exten

sion (I) yields the measure space ( [0 , 1] , S), , )') and the (Lebesgue)

integral 11 . d)'(x) . The function f in 2.2.53 is Lebesgue integrable but not Riemann integrable.]


2.2.56 Exercise. If f E L l ( [0, 1] , ),) and E > 0: a) for some positive M, i f AM �f E« lf l , M), then ), ( [0, 1] \ A) < E and

b) for some lsc h, h 2': f and 11 h - f d)' (x) < E; c) for some usc k, k � f

and 11 f - k d),(x ) < E.

2.2.57 Exercise. For a set X and a function lattice L contained in JR.x , the smallest monotone class of functions containing L, i .e . , the Baire space B(L) associated to L, is a function lattice.

[Hint: For f in B( L) if M (f) consists of all g such that

{g, f + g, f V g, f 1\ g} c B(L) , M(f) is a monotone class.]

2.2.58 Exercise. The set function

if XE E Ll otherwise

is a measure and if f E LI , then J (f) = Ix f (x) df.-lo (x) . Furthermore, if f.-ll : D '3 E r-+ f.-ll (E) E [0, 00] is a measure and for all f in L l ,

J(f) = Ix f(x) df.-ll (X) ,

then for all E in D, f.-lo (E) � f.-ll (E) .

2.2.59 Exercise. a) The set R([O, I] of functions f that are Riemann integrable on [0, 1] is a function lattice. b) If X �f [0, 1] ' L �f e( [O, 1 ] , JR.) ,

and when f E L, 1(f) �f 11 f(x) dx (Riemann integral) , then L e R e Ll .

c) If R([O, 1] ) ::) {fn }nEN and fn + 0, then 11 fn(x) dx + o.

[Hint: If f E R([O, 1] ) , for a sequence {sn }nEN of step-functions , Sn t f a.e. (),) . If s is a step function, there is a sequence {cn }nEN of continuous functions such that Cn t s.]


2.2.60 Exercise. For the Dirac functional rSa in 2 .1 .21 , what are D, D , and the corresponding measure?

2.3. The Riesz Representation Theorem

An important consequence of the preceding development is the following theorem of F. Riesz. The result applies, when X is a locally compact Hausdorff space, to nonnegative linear functionals defined on Coo (X, JR.) , the set of continuous functions f such that supp (I) is compact . In Chapter 4 the result is extended for continuous C-valued linear functionals defined on Coo(X, C) .

For a locally compact Hausdorff space X, the class of Baire sets Sb (X ) is the intersection of all a-algebras contained in !fj(X) and including, for each f in Coo (X, JR.) and each a in JR., the set Edf, a) . Hence if 0 denotes any one of � , >, 2: and a E JR., then Sb (X) contains Eo (I, a ) . However,

is a compact Gli , whence Sb (X) is the a-algebra generated by the set of compact GliS.

2.3.1 Example. If #(X) > No, O(X) �f !fj(X) (the topology of X is the discrete topology) , and X* �f Xl.,J{y} is the one-point compactification of X (cf. 1. 7.28) , each neighborhood of y is closed (and, by definition, open) . Each finite subset of X is a compact Gli , whereas the set {y} is compact but is not a Gli . An infinite subset A of X is not compact , since its elements regarded as sets constitute an open cover V of A, and no finite sub cover of V exists. Thus the a-algebra SG8 generated by the compact Glis of X* is the a-algebra Sf (X) generated (in X* ) by the finite subsets of X. The construction given in 2.2 .40 for S,B implies, in the current instance, that {y} tic Sf (X) . Thus S,B (X* ) "1-Sb (X* ) : the sets S,B and Sb need not be the same.

2.3.2 THEOREM. (F. Riesz) FOR A LOCALLY COMPACT HAUSDORFF SPACE X AND A NONNEGATIVE LINEAR FUNCTIONAL

1 : Coo (X, JR.) '3 f r-+ 1(1) E JR.,

FOR SOME MEASURE SPACE (X, Sb, p,) AND ALL f IN Coo (X, JR.) ,

1(1) = Ix f(x) dp,(x) .

Section 2.3. The Riesz Representation Theorem 77

PROOF. If Coo (X, JR.) '3 In + 0 , since supp UI ) is compact and

supp UI ) ::) supp Un ) , n E N,

Dini's Theorem (1 .2 .46) implies In � O. Urysohn's Lemma (1 .2 .41) implies there is in Coo (X, JR.) a g such that 0 :::; g :::; 1 and g 2': I on supp (II ) .

If m E N, for some no , 0 :::; In (x) < !!.-. if n > no , whence m

1 Un ) :::; 1 (�) = �1(g) . Hence lim 1 Un ) = O. Since 1 Un ) 2': 1 Un+ I ) , 1 Un) + 0: 1 is a DLS func-n---+= tional.

By abuse of notation, 1 /\ Coo (X, JR.) c Coo (X, JR.) , whence [2.2.29d)] the set D of DLS measurable sets is a complete a-algebra that includes, for each I in Coo (X, JR.) and each a in JR., Edl, a) : D ::) Sb(X) . D

2.3.3 Exercise. If X is a locally compact Hausdorff space,

K E K(X) , and K - -n- l, l = On E O(X) ; b) On E K(X) ; c)

K c g� u, 1 ) = n On · In sum, every compact set K is contained in a nEN

compact Cij . [ 2.3.4 Remark. When X is a locally compact space, KGij is the set of all compact sets each of which is a Cij . Some define Sb(X) to be the a-ring generated by KGij : Sb(X) �f aR (KGij) and some define S,(3 (X) to be aR[K(X)] . The reader is encouraged to explore the relations among these definitions and those used in this book.]

2.3.5 Example. In 2 .1 .22, if X �f JR., D consists of all functions I such that supp U) is empty, finite, or countable while

{J E Ll } {} { U E D} /\ {� II(X) I < 00 } } . Consequently, JR. tt D.

The function I : JR. '3 x r-+ x E JR. is such that # [supp U)] > No, whence I tt D, although for each a in JR., E=U, a) = {a} E D (cf. 2 .2 .22) . 2 .3 .6 Example. In its customary topology, again JR. is a locally compact Hausdorff space. If I E Coo(JR., JR.) and 1U) �f l I(x) dx (the Riemann integral of I) , Urysohn's Lemma implies that aR[K(JR.)] = D.


2.3.7 Exercise. In 2.3.2: a) if E E 5,6,

p,(E) = inf { p,(U) : E c U, U E O(X) } ;

b) if p,(E) < 00, then p,(E) = sup { p,(K) : K c E, K E K(X) } . The conclusions a) and b) above motivate the following terminology.

2.3.8 DEFINITION . FOR A MEASURE SPACE (X, 5 , p,) , A SET E IN 5 IS outer regular ( inner regular) IFF

p,(E) = inf { p,(U) : E C U E O(X) } , (p,(E) = sup { p,(K) : E ::J K E K(X) }) .

AN E THAT I S BOTH OUTER REGULAR AND INNER REGULAR I S regular. WHEN EVERY E IN 5 IS OUTER REGULAR ( INNER REGULAR) (REGULAR) , (X, 5, p,) AND P, ARE OUTER REGULAR ( INNER REGULAR) (REGULAR) .

2.3.9 Exercise. In 2.3.2, if X �f [0, 1] in its customary topology and I is the Riemann integral r : a) p, = A (cf. 2.2.40) ; b) if E E 5,6( [0, 1] ) and

ira . l ] x E JR., then

clef X + E = { x + Y : Y E E } E 5,6( [0 , 1] ) , A(E) = A (X + E)

(5,6 ( [0, 1] ) and A are translation-invariant) ; c ) if

{E, F} c 5,6 , ° < A(E) . A(F) < 00, and x E JR.,

then f (x) � A[E n (x + F) ] = l XF(Y - x)XE(Y) dy and f is continuous;

d) E - F �f { x - y : x E E, y E F } , contains a neighborhood of zero. [Hint: If E is an interval, b) and c) are valid. For d) , c) applies.]

[ 2.3.10 Note. In the context of 2.3.9, the measure space is [JR., 5), (JR.) ) , A] and A is Lebesgue measure . The sets in 5), [JR.)] are the Lebesgue measurable sets . Corresponding definitions apply for the notions of Lebesgue measurable functions, Lebesgue integrable functions, Lebesgue integrals , etc.]

2.3 .11 Exercise. (Vitali-Caratheodory) In the context and notation of 2.3.2 , the conclusion in 2.2.56 obtains.

[Hint: Urysohn's Lemma (1.2.41 ) applies.]

Section 2.4. Complex-valued Functions 79

2.4. Complex-valued Functions

Little difficulty and much advantage follow from admitting C-valued functions to the discussion. From this point forward, functions with numerical ranges are to be assumed as C-valued unless the contrary is stated.

The image "(* �f "((lR.) of the curve

1 - t2 . 2t "( : lR. '3 t r-+ 1 + t2 + l 1 + t2 E C

is { z I z l = 1 , z i- -I } �f '][' \ {- I } . The function

clef it 2 fJ : lR. '3 t r-+ fJ(t) = --2 dx o 1 + X

(2.4.1 )

(2 .4.2)

is continuous and strictly monotonely increasing. For some (finite) number, denoted 7r, lim fJ( t) = ±7r: "( is rectifiable and

t---+±oo

length of "( �f £("() = 27r .

The inverse of fJ is the function t : (-7r, 7r) '3 fJ r-+ t(fJ) �E (-00, (0) :

t (O) = 0, lim t (fJ) = 00, lim t (fJ) = -00, t(±7r) �f ±oo. Ot7r O.j.-7r

The two trigonometric functions,

1 - t(fJ) 2 cos : [-7r, 7r] '3 fJ r-+ -----'--'-::-1 + t(fJ) 2 . 2t(fJ) sm : [-7r, 7r] '3 fJ r-+ 1 + t(fJ) 2 '

are infinitely differentiable on (-7r, 7r) and

The formal

cos' fJ = - sin fJ, sin' fJ = cos fJ.

fJ2 fJ4 1 - - + - - · · ·

2! 4! ' fJ3 fJ5

fJ - - + - - · · · 3! 5!

(2.4.3)

(2.4.4)

(2.4.5)

(2 .4 .6)

(2 .4 .7)

converge for all fJ in C. The remainder formulCE associated with the Maclaurin polynomials for cos fJ and sin fJ show that (2.4 .6) resp. (2 .4 .7) represent


cos fJ resp. sin fJ on (-7r, 7r) and define cos fJ and sin fJ throughout C. Combined, they yield

cos fJ + i sin fJ = 1 + � (ifJ)n �f exp( ifJ) (Euler 's formula) � n!

and the definition n=l

clef 00 fJn exp(fJ) = 1 + L ,.

n. n=l (2.4.8)

The right member of (2.4.8) converges throughout C. On JR., the series (2.4.6)-(2.4 .8) represent three infinitely differentiable functions mapping JR. into JR.. (By virtue of the argument in 5.3.2, they are infinitely differentiable throughout C.) Direct calculations using (2.4.8) show

exp(u + v ) = exp(u) exp(v) , (2.4.9)

whence if e � exp(I ) , successively,

exp(n) = en , n E N, exp(m) = em , m E Z, exp r = eT, r E Q. (2.4. 10)

Owing to the continuity of exp, the definition eO �f exp( fJ) , fJ E C, is consistent with the formulre in (2.4 . 10) .

1 If fJ E JR., I eiO I = (cos2 fJ + sin2 fJ) "2 = 1 . By virtue of (2.4.3) ,

e 7ri = cos 7r + i sin 7r = -1, (2.4. 1 1 )

k i { COS(fJ + 2k7r) } { cos fJ } whence, for k in Z, e2 7r = 1 . If fJ E C, then . ( k ) = . . sm fJ + 2 7r sm fJ

If ¢ E JR., for a unique k in Z, fJ � ¢ + 2k7r E (-7r, 7rJ . If cos ¢ = 1, then cos fJ = 1, i.e., t (fJ) = fJ = 0, ¢ E 2Z7r . If Z � x + iy and eZ = 1, then x = 0 and eiy = 1 whence eZ = 1 iff Z E 2Z7ri . (The last conclusion is alternatively deducible from the formula

cos fJ = 1 _ (fJ2 (1 _ fJ2 ) + fJ6 (1 _ fJ2 ) + . . . )

2 12 6! 56

applied when fJ E [-7r, 7rJ . ) 2 .4.12 Exercise. The least positive period of both cos and sin is 27r.

[Hint: sin fJ = - cos' fJ.J

2 .4. 13 Exercise. a) exp llR is a strictly monotonely increasing function; b) the function inverse to the exponential function exp on JR. is the logarithmic function In : (0, (0) '3 Y r-+ In(y) E JR., i.e. ,

exp 0 In : (0, 00) '3 Y r-+ y, In 0 exp : JR. '3 x r-+ x;

Section 2.4. Complex-valued Functions

c) lim exp(x) = 00, lim exp(x) = 0; d) if z = x + iy, X400 X4-00

eZ = eX (cos y + i sin y) ;

81

e) for z in C \ {a} and some unique fJ in ( -7r, 7r] , Z = exp(ln I z l + ifJ) ; e) exp' = expo

[ 2.4. 14 Note. For ,,( as in (2.4 . 1 ) , if z �f x + iy E "(*

, there is in (-7r, 7r) a unique 8(z) such that ei8(z) = z . Hence, if e<l> = z, then e<l>-i8(z) = 1 and so for some k in Z, ¢ = i [8(z) + 2k7r] .

The signum function sgn is defined by:

cl f -{ I z l sgn : C '3 z r-+ sgn (z) �

OZ if z :;to O

otherwise

Hence: a) z · sgn (z) = I z l ; b) sgn is continuous on C \ {O}; c)

I ( ) 1 { I if z :;to O sgn z = o otherwise.

The unit circle { z : I z l = I } (= "(* u {- I} ) is the same as 'lI'.

If z E (C \ {O} and z = Iz l eiiJ , -7r < fJ � 7r, the half-line clef { ·0 [0 = w : w = re' , r 2: 0 } meets 'lI' in exactly one point , which is sgn (z) . If U is open in C, u n 'lI' is relatively open in 'lI' and

sgn - 1 (U) = S = z { clef { S U {O}

[0 n U n 'lI' :;to (/) } if 0 tic U otherwise

Hence sgn - 1 (U) is either open or the union of an open set and a single point : sgn is (aR[O(C)] , aR [O(C) )] -measurable. When z �f x + iy and sgn (z) �f u(x, y) + iv(x, y) , then u, v E D.

For X, L, and I in the DLS development , a function

f : X '3 x r-+ f(x) = SRf (x) + i':Sf(x) �f u(x) + iv (x) E C

is defined to be DLS measurable, D-measurable, or Caratheodory measurable iff u and v are: f E LI iff both {u, v } e Ll , in which event ,

I l f l l l �f J( lf l ) .


For a measure space (X, S , I-l) , by definition,

{I �f u + iv E S} {} { { u, v } C S} ,

{J E L1 (X, I-l) } {} { I I I E Ll (X, I-l) } . In the circumstances, smce ma:x{ lu l , I v l } � ';u2 + v2 � lu i + l v i , I E Ll resp. I E L1 (X, I-l) iff

J( l u l ) + J( lv l ) < 00 resp. lx , u , dl-l + lx , v , dl-l < 00.

Unless the contrary is stated, henceforth, functions will be assumed to be C-valued. 2.4. 15 Exercise. The map JR. '3 x r-+ exp(27rix) E 'lI' is a continuous open epimorphism of the additive group JR. onto the multiplicative group 'lI'. (In particular, if k E Z, then exp((2k + 1)7ri) = -1. )

2 .4.16 Exercise. If z i- 0: a) Arg (z) �f { (J : (J E JR., z = I z leiO } i- 0; b)

{ . . (Jl - (J2 # [Arg (z)] = No ; c) (Jl , (J2 } C Arg (z) Imphes E Z. 27r 2.4. 17 Exercise. If -7r < (J � 7r: a) cP : C \ [0 '3 z r-+ B[sgn (z)] is a nonconstant continuous map; b) cP (C \ (0 ) c U Arg (w) .

wEC\(e

2.4. 18 THEOREM. a) IF THE CURVE "( : [0, 1] '3 t r-+ "((t) E C IS NONCONSTANT AND 0 tt "(* , FOR SOME NONCONSTANT cP,

cP o "( : [0, 1 ) '3 t r-+ U Arg (w - o) wey'

IS CONTINUOUS . b) A MAP '1jJ DEFINED ON ,,([ [0, 1) ] AND SATISFYING

'1jJ 0 "( : [0, 1 ) '3 t r-+ U Arg (w - 0) wE"!'

IS CONTINUOUS IFF FOR SOME m IN Z, '1jJ 0 "( - cP 0 "( = 2nm.

(2.4. 19)

PROOF. a) For the curve ;y �f "( - 0, if ° = to < tl < . . . < tn = 1 and

is sufficiently small (and positive) , the arc ;Y [tk- l , tk] is contained in some C \ [0 where 2 .4 .17 applies: there is a cPk such that cPk o ;Y is continuous on [tk- l , tk] . Owing to 2.4. 16c) ,

for some ml in Z, cPl o ;Y (t t ) = cP2 o ;y (t t ) + 27rm l , for some m2 in Z, c/J2 o ;Y (t2 ) = cP3 o ;y (t2 ) + 27rm2 ,

Section 2.4. Complex-valued Functions

etc. Thus if

then 1> is continuous on [0, 1 ) and if t E [tk- l , tk ) , then

if t E [0, t I ) if t E [t l , t2 )

if t E [tn- I , 1 )

Since the correspondence ;Y(t) +-+ l'(t) is bijective, the equation

defines the required ¢.

83

b) If m E Z, then 27rm + ¢ 0 l' is continuous on [0, 1) and satisfies (2.4 .19) .

Conversely, since the map '1jJ 0 l' - ¢ 0 l' is continuous, the result in 2.4. 16c) implies that '1jJ 0 l' - ¢ 0 l' is 27rZ-valued. Hence '1jJ 0 l' - ¢ 0 l' is a constant . [] 2.4.20 Exercise. In the context of 2.4.18: a) If r5 is sufficiently small (and positive) , then ¢ 0 1' is monotone on each interval [tk- l , tk ) ' b) For A clef . ( ) . d ( ) clef A - ¢ 0 1'(0) . d .

= hm ¢ 0 l' t , Ill 'Y a = , the m ex of l' wzth respect to a, �l 27r

is in Z. c) On each component of C \ 1'* , ind 'Y( a) is a continuous (hence constant) function of a. d) Only one component of C \ 1'* is unbounded. e) If a lies in the unbounded component of C \ 1'* , then ind 'Y (a) = 0.

[Hint: e) If E > ° and 10' 1 is large enough, then in the notations of 2.4.18, sup l1> o ;y I < E.]

tE [a , l ) When z E C \ {O} , the count ably infinite set In( l z l ) + Arg (z) is de

noted Ln (z) . 2.4.21 Exercise. a) If 1' : [0, 1] 3 t r-+ l'(t) E C is continuous and ° tt 1'* , for some ,£ defined on 1'[ [0, 1 ) ] , ,£ 0 l' : (0, 1 ) 3 t r-+ U Ln (w ) is contin-wE'Y' uous. b) If z i- ° and eW = z, then w E Ln (z) . c) If t in [0, 1 ) , then e£o'Y(t) = l'(t) .

[Hint: The argument in the proof of 2.4.18 applies.] There are profound connections between the map h, a} r-+ ind 'Y(a)

and basic topology, e.g. , the Jordan Curve Theorem, Brouwer degree of a map, etc. A dense but useful reference here is [Sp] where an extensive


bibliography is offered. The discussion provided above for the basic topics of this subject is adequate for the current and later purposes of this book.

2.5 . Miscellaneous Exercises

An element A in a a-ring 5 is defined to be indivisible iff every proper 5-subset B of A is 0, i.e. , { 5 '3 B ¥A} ::::} {B = 0} .

2 .5.1 Exercise. a) If the set I of indivisible elements of a-ring 5 is finite and U is their union, 5 �f { A \ U : A E 5 } is a a-ring and no element of 5 is indivisible. b) If I is infinite, #(5) > No . c) If 5 is infinite, then 5 is infinite and #(5) > No . In sum, if 5 is a a-ring, then #(5) i- No .

[Hint: If 5 '3 A i- 0, some nonempty S-set B is a proper subset of A.]

2 .5 .2 Exercise. For a group G, a function lattice L contained in JR.G , a DLS functional 1 : L r-+ JR., an I in JR.G , and an a in G, if the left atranslate of I is I[a] : G '3 x r-+ I(ax) , and for each a in G and each I in L, 1 (f[a] ) = 1(f) , then: a) for all I in D, I[a] E D; b)

{f E L1 } ::::} { {f[a] E L1 } !\ {J (f[a] ) = J(f) } } ;

c) {E E D} ::::} {{aE E D} !\ {t-t(aE) = I-l(E) }} (cf. 2 .3.9) . 2.5.3 Exercise. If I E L1 (X, I-l) and lim I-l (En ) = 0, then n---+=

lim r il l dl-l = o. n--+CXJ } En [Hint: If I is a simple function the result is a consequence of the nonnegativity of I-l. For the general case the density of the set of simple functions in L1 (X, I-l) applies.]

2 .5.4 Exercise. If n : A '3 ), r-+ n(),) E !fj(X) is a net, then

lim X ( ' ) = x- and lim X ( ' ) = X . .AEA n A n .AEA n A !l

2 .5.5 Exercise. An I in JR.[O. 1 ] Riemann integrable iff: a) for some finite M and all x, I I (x) I � M and b) the Lebesgue measure of the set Discont (f) of discontinuities of I is zero: ), [Discont (f)] = O. 2.5.6 Exercise. If 5 is a a-algebra contained in !fj(X) ,

h : JR.2 '3 (x, y) r-+ h(x, y) E JR.

Section 2.5. Miscellaneous Exercises 85

is continuous, and Ii : X '3 x r-+ Ji(x) E JR., i = 1 , 2 , are S-measurable, then H �f h (II , h ) is S-measurable.

[Hint: a) The set E< (h, a) is an open subset U of JR.2 ; b) E< (H, a) consists of all x such that [II (x) , h (x)] E U; c) U is a union of (count ably many) pairwise disjoint half-open rectangles

2.5.7 Exercise. If {1 , J} C D, then sgn (I) E D.

[Hint: The set E �f E= (I, 0) is in 5 and I (X \ E) c C \ {O} ; C \ {O} '3 w r-+ sgn (w ) is continuous, whence sgn (I) is measurable on X \ E.]

2.5.8 Exercise. If 5 is a a-ring contained in !fj(X) , S is the associated set of S-measurable functions, and I E S, there is in S a (J such that I(x) == I I(x) l eiO(x) .

2.5.9 Exercise. a) If 5 is a a-ring contained in !fj(X), S is the associated set of S-measurable functions, and I E S, then I I I E S. b) The converse of a) is false. c) {J E L 1 } {} {{J E S} !\ { I I I E L 1 } } .

[Hint: For b) , if E E !fj(X) \ S,(3 (JR.) , then IXE - X(lR\E) I == 1 . ]

2.5.10 Exercise. What is the result of applying the DLS procedure to the lattice L �f L1 (X, p,) and the functional I : L '3 I r-+ Ix I dp,?

2.5. 11 Exercise. If R is a ring of sets and is monotone, i.e. ,

{ { {En}nEN C R } !\ {En C En+d } '* { U En E R} , nEN

{ { {En }nEN C R} !\ {En ::J En+d } '* { n EN E R} , nEN

then R is a a-ring.

2.5.12 Exercise. For a a-ring 5, if {In}nEN is contained in the corresponding set S of S-measurable functions, each of lim In , lim In , and

n --+ CXJ n --+ CXJ (when it exists) lim In is in S. n---+= 2.5.13 Exercise. If: a) (X, S , p,) is a measure space; b) X is totally finite , i .e. , X E S and p,(X) is finite; c) p,* is the induced outer measure; d)


(X, C, ll) is the measure space for the a-algebra of Caratheodory measurable sets, then C = 5 (the completion of S ) .

[Hint: If E c F E 5 and p,(F) = 0, then p,* (E) = 0: 5 c C. If A E C, then for sequences {An}nEN and {Bn}nEN contained in S , E C An ::) An+l , n E N, E ::) Bn C Bn+l , n E N, and

fI(A) = p,* (A) = lim p, (An ) = lim p, (Bn) .] n --+ ex) n --+ CXJ

2.5.14 Exercise. In 2.5 .13 the conclusion remains valid if X is the countable union of sets of finite measure, i.e. , if X is totally a-finite.

2.5.15 Exercise. For a curve "( : [0, 1] '3 t r-+ "((t) E C and an a not in "(* : a) for some positive J, inf h(t) - 0'1 2: J and o:s;t:S; 1

b) if 0 = t l < t2 < . . . < tn = 1 , sup tk - tk- l < J, and 2:S;k:S;n

n then for some m in Z, e � 2)'h = 2m7r; c) ind ')'(O') = m.

k=2

2.5.16 Exercise. If f E JRlR, fOR.) C [-00, (0) , and f is usc: a) f is Lebesgue measurable; b) A(E) < 00 implies either Ie f(x) dx E lR. or, by

abuse of notation, Ie f(x) dx = -00. Corresponding statements are valid when f(lR.) C (-00, 00] . 2 .5 .17 Exercise. If X is a set and {En}nEN C !fJ(X) , then: a) There are

-1· - E clef E 1· E clef E h th sets 1m n = resp. 1m n = _ suc at

b)

n---+= n---+oo

lim XE = XE and lim XE = XE. n--+CXJ n n--+CXJ n -

= lim En = { x X is in infinitely many En } = n U En ,

mEN n=m =

X is in all but finitely many En } = U n En · mEN n=m

Section 2.5. Miscellaneous Exercises

c) lim En C lim En . d) If 5 is a a-ring and {En }nEN C 5, then n --+ CXJ n --+ CXJ

lim En E 5 and lim En E 5. n---+=

87

2.5.18 Exercise. If (X, 5, p,) is a measure space and £ is the set of simple functions s such that Ix s dp, E JR., how does the completion of £ with re-

spect to the metric J : £2 '3 {J, g} r-+ J(f, g) �f Ix I f - g l dp, compare with

Ll (X, p,)? 2.5 .19 Exercise. (Egorov) If: a) (X, 5 , p,) is totally finite; b)

c) for each x, fn (x) -+ f(x); and d) E > 0, there is in 5 a set E such that p,(X \ E) < E and fn l E� fi E (v. [GeO] ) .

For each x in JR., lim X r ] ) (x) = 0 yet i f 00 > A(E) > 0 , on JR. \ E, n---+= ( n,n+l then X( r ] ) fi o. n,n+l

[Hint: If ENm �f n9

N { x : Ifn (x) - f(x) 1 2: � } , for large N,

P, (ENm) < ETm .] The symmetric difference Al1B of two sets is (A \ B) U (B \ A) .

2.5.20 Exercise. If (X, 5, p,) is a measure space: a) 5 is closed with respect to the formation of symmetric differences; b) for elements A and B of 5, the relation {A rv B} {} {p,(Al1B) = O} is an equivalence relation.

The rv-equivalence class containing A is denoted A� .

2.5.21 Exercise. For rv as in 2.5.20, the set 5� �f 51 rv of equivalence classes, the map

is well-defined, i.e. , independent of the choice of the representatives A resp. B of A� resp. B� . Furthermore, p is a metric in 5� . 2.5.22 Exercise. In the context of 2.5.20 and 2.5.21 , if

X �f JR., 5 �f 5,(3 (JR.) , P, �f A ,

then (5-; p) is not a complete metric space. Its p-completion is D, i.e. , 5-is p-dense in 5.


2.5.23 Exercise. If X is a set, S C !fj(X) , and

M �f { Sa : Sa C S, # (Sa ) � No } ,

then aR(S) = U aR (Sa ) .

[Hint: The right member of the preceding equation is a a-ring.]

2.5.24 Exercise. If E E S), (ffi.) , then I : ffi. '3 t r-+ I(t) � >. ( [0, t) n E) is continuous. The previous assertion is valid if [0, t) is replaced, for any a in [-00, (0) by any of [a , t) , (a , t ) , (a , t] , [a, t] , or by any of the last four when a and t are interchanged. If g : ffi. '3 t r-+ g( t) E ffi. is Lebesgue measurable and t in the first sentence is replaced by g(t) , is I : t r-+ >. { [a, g(t) ] n E} Lebesgue measurable? 2.5.25 Exercise. a) For measure spaces (X, S, f.-ln ) , n E N, such that f.-ln � f.-ln+l , f.-l � sup f.-ln is a measure. b) For (ffi., S)" >.) if f.-ln �f .!>., n E N, n n h clef . f . t en f.-l = m f.-ln IS not a measure. n

2.5.26 Exercise. If I E (Lu n ffi.X) , for some nonnegative p in Lu and some II in L, 1 = p + II ·

00 [Hint: If L '3 In t I, L Un - In- I) E Lu .]

n=2

2.5.27 Exercise. For some sequence {In}nEN contained m Ll (ffi., >.) ,

In � 0, while l in d>' t 00 .

2.5.28 Exercise. If En �f [n, (0) , n E N, then: • S), '3 En ::) En+1 ; . · n En = 0;

nEN • >. (En ) == 00 ;

cf. 2 .2.26.

3 Functional Analysis

3.1. Introduction

For a set X, there are various important subsets of CX , e.g., L 1 , L 1 (X, p,) , C( X, JR.) , etc. Each of these is an JR.-vector space or a C-vector space and is endowed with a topology related to its manner of definition. Thus L 1 and L 1 (X, p,) are metric spaces, whereas C (X, JR.) inherits a topology from CX viewed as a Cartesian product. In short, each is a paradigm for a topological vector space .

3.1 . 1 DEFINITION. A topological vector space (TVS) (V, T) (OR SIMPLY V) IS A C-VECTOR SPACE ENDOWED WITH A HAUSDORFF TOPOLOGY T SUCH THAT THE MAPS

V x V '3 (x, y) r-+ x + y E V, C x V '3 (z, x) r-+ zx E V,

FOR VECTOR ADDITION AND MULTIPLICATION OF VECTORS BY SCALARS (ELEMENTS OF C) ARE CONTINUOUS . THE ORIGIN (THE ADDITIVE IDENTITY) OF V IS DENOTED O. WHEN SOME NEIGHBORHOOD BASE FOR T CONSISTS OF CONVEX SETS , V IS A locally convex topological vector space (LCTVS ) .

The class of locally convex topological vector spaces includes the class of normed spaces , i.e. , the class of those vector spaces V for which there is a norm, namely a map I I I I : V '3 x r-+ I lx l l E [0, (0) such that: a) I lx l l = 0 iff x = 0; b) I lx + y l l � I lx l l + I ly l l ; c ) for z in C and x in V, I l zx l l = I z l . I lx l l .

3 .1 .2 Exercise. If (V, I I I I ) is a normed space, then

I lx - y l l 2': I l lx l l - I ly l l l ·

When (V, I I I I ) is a normed space, d : V2 '3 (x, y) r-+ I l x - y l l is a metric for V. When (V, d) is complete, V is a Banach space .

THEOREM 2.2.32 implies L1 and, for any measure space (X, S, p,) , L1 (X, p,) are Banach spaces.

89

90 Chapter 3. Functional Analysis

When 1 < p < 00, the set LP resp. LP(X, p,) consists of the DLS measurable resp. S-measurable C-valued functions I such that

In Section 3.2 it is shown that if 1 < p < 00 and, according to the convention adopted, each null function is regarded as 0, I I l i p i s a true norm and that LP and LP(X, p,) are complete with respect to I I l i p : each is a Banach space.

For a topological space X, CX contains:

a) Coo (X, q , the set of continuous functions I for which supp (f) is compact (v. 1 .7.23) ;

b) Co (X, q consisting of those continuous functions I such that for each positive E, K.(f) �f { x : I I(x) 1 2: E } is compact .

For I in Coo (X, q or in Co(X, q , 1 1 1 1 100 � sup II (x) 1 < 00. xEX

3.1 .3 THEOREM. WITH RESPECT TO I I 1 1 00 , Co(X, q IS A BANACH SPACE.

PROOF. The verification of the norm properties a)-c) for I I 1 1 00 is straightforward.

If {fn}nEN is a Cauchy sequence in Co(X, q , then for each x in X, {fn (x)}nEN is a Cauchy sequence (in q , whence I(x) �f lim In (x) exists. n---+oo If E > 0, since II l loo-convergence is uniform convergence, for some N and all x, {m , n > N} ::::} { l lm (x) - In (x) 1 < E} , whence

lim Ilm (x) - In (x) 1 = I I (x) - In (x) 1 � E. m---+oo

In short, In � I, i.e. , lim I I I - In l l oo = o. (The preceding argument is n---+oo valid as well if {fn}nEN is a Cauchy sequence in Coo(X, q : for some I, In � I· However, as shown in 3. 1.5, Coo (lR., JR.) is not II l lao-complete. ) def If E > 0, S. = { x : I I(x) 1 2: E } , and m 2: 2, then for some nm and all n greater than nrn , Sc C { x : I ln (x) l 2: (1 - �) E } �f Knm. By defi-

nition, each Knm is compact and S. C n n Knm � K, which is also

compact . On the other hand, if x E K, m 2: 2, and n 2: nm , then

Section 3.1 . Introduction 91

whence I I(x) 1 2': (1 - �) E and x E Sf : Sf = K. Thus I E Co(X, q .

D

[ 3 .1 .4 Note. When X is compact,

Coo (X, q = Co(X, q = C(X, q.]

def � sin 27rx 3.1.5 Example. If X = JR. and In(x) = � X[k,k+l] . k ' n E N, then k=l 00

{ U ""' sin 27rX def In}nEN c Coo (X, q and In (x) -+ � X[k,k+ l] . k = I(x) although k=l I tt Coo (X, q: Coo( X, q need not be a Banach space with respect to the

norm I I 1 100 ' 3 .1 .6 Example. The LCTVS Coo (JR., q is a l i l l I -dense subset of Ll (JR., >.) and Coo (JR., q ¥Ll (JR., >.) . Hence Coo(JR., q is not l i l l I -complete.

The following construction, of independent interest, validates the preceding statements and provides an explicit II I l l -Cauchy sequence contained in Coo (JR., q and for which the I I I I I -limit i s in L 1 (JR., >.) \ Coo (JR., q .

For n in N,

In : JR. 3 X f-t

1

nx n(1 - x) o

1 1 if - < x < 1 - -n - - n

'f 1 1 0 < X < -- n

1 if l - - < x < 1

n -otherwise

is in Coo (JR., q and In 1 1�1 X[ ] . If a < b, there are real constants a, (3 such 0, 1 that if gn (x) �f In (ax + (3) , then gn 1 1�1 X[ ] . It follows that Coo (JR., q is a,b I I I l l -dense in Ll (JR., >.) .

For a in (0, 1 ) and an enumeration {hhEN of the intervals deleted in the construction of the Cantor set Co: (v . 2 .2 .40) there are real con-def --stants ak , (3k such that if Ink (X) = In (akx + (3k ) , then supp (Ink ) = h 00 and Ink 1 1�1 Xh ' If gn �f Link , then {gn}nEN is a I I I l l -Cauchy sequence

k=l contained in Coo (JR., q and if its I I I l l -limit is g, then X[O, l ] - g = X(Ca ) is not a null function and is not in Coo (x, q .


3.2. The Spaces LP , 1 � p � 00

Henceforth, L 1 denotes some L 1 (X, p,) or some L 1 derived from a DLS functional I. As noted earlier , Ll (X, p,) and Ll are, for appropriately related p, and I, essentially the same. Similarly S and D differ only by a set of null functions. For p in [ 1 , (0) , LP �f { I : I E S, I I IP E LI } and, when I E LP, I I I I I � = I I I I IP I l l '

3 .2 .1 LEMMA . (Young) IF : a) ¢ IS A STRICTLY MONOTONELY INCREASING CONTINUOUS FUNCTION DEFINED ON [0, (0) ; b) ¢(O) = 0; c) '1jJ �f ¢- 1 ; d)

<t>(x) � lx ¢(t) dt, AND \II(y) �f lY '1jJ (s) ds;

AND e) {a , b} C [0, (0) ; THEN ab � <t>(a) + \II (b) . EQUALITY HOLDS IFF b = ¢(a) .

PROOF. In the context , the roles of ¢ and '1jJ resp. <t> and \Ii resp. a and b are symmetric. Hence it may be assumed that ¢( a) � b. The geometry of the situation in Figure 3.2 .1 implies that the rectangle [0, a] x [0, b] is contained in

{ (x , y) : x E [0, a] , 0 � y � ¢(x) } U { (x, y) : y E [0, b] , 0 � x � '1jJ (y) } ,

whence ab � <t>(A) + \II (b) . Equality holds iff b = ¢(a ) . D 1 1 For p in ( 1 , (0) , there is in '( I , (0) a unique p' such that - + - = 1 : p p'

p' = -p-. The numbers p, p' form a conjugate pair . p - 1

y - axis I I (a, <p (a» I

Figure 3.2 .1 .

Section 3.2. The Spaces LP, 1 :::; p :::; 00 93

3.2.2 COROLLARY. IF p E ( 1 , 00 ) , f E LP , 9 E LP' , THEN f g E L 1 AND I l fg l l l :::; I l f l lp . I l g l l p" (HOLDER'S INEQUALITY)

PROOF. The measurability of fg is a consequence of 2 .5 .6. If ¢(x) �f xp- 1 , 1 xP xp'

then '1jJ(x) = ¢- I (X) = x p- 1 and <t>(x) = - , W(x) = - . Thus 3 .2 .1 im-p p'

plies

I f(x)g(x) 1 :::; I f(xW + Ig(xW' p p'

If I l f l l p = I l g l l p' = 1 , integration of both members of (3.2.3) yields

(3.2.3)

If either f or 9 is 0, the conclusion is automatic. If neither f nor 9 is 0,

h 1 F clef f clef 9 . . t ey may be rep aced by = -1 -1- resp. G = -1 1 -1 1- ' m whIch case the I f Ip 9 P' previous argument applies to F and G. D

3.2.4 COROLLARY. EQUALITY IN HOLDER'S INEQUALITY OBTAINS IFF

I I f l i p . I lg l l p' = 0 OR I l f l l p . I l g l l p' i' 0 AND I:j:�� I :::�;; '

PROOF. Trivialities aside, if, for all x in a set of positive measure,

for some n in N and all x in a set E of positive measure,

Owing to the criterion for equality in Young's inequality (3.2 .1 ) , for some

positive E and all x in E, I f(x) 1 ' lg(x) 1 < I f (xW + Ig (xW'. Integration

I l f l l p ' I l g l l p' I l f l l p I lg l l p' over E of both members of the inequality above yields

D

3.2.5 COROLLARY. IF f, g E LP , THEN f + 9 E LP AND

I l f + g l lp :::; I l f l lp + I lg l lp ( MINKOWSKI 'S INEQUALITY) .

EQUALITY OBTAINS IFF FOR SOME NONNEGATIVE CONSTANTS A , B , NOT BOTH ZERO, Af � Bg.


PROOF. Since I f IP, Ig lP E Ll , a vector lattice, h �f I f lP V I l g l P E Ll , whence I f + glP :::; 2h E LI . Since I f + glP :::; I f + glP- l . I f I + If + g lP- l . Ig l , integration of both members of the last inequality, the identities relating p

.E. and p', and Holder 's inequality imply I l f + g l l � :::; I l f + g i l;' ( I I f l i p + I lg l l p ) ·

.E. Division by I l f + g i l;' leads to Minkowski's inequality.

If equality obtains, I f + g lP == I f + g IP- l ( l f l + Ig l ) , i.e. ,

I f + g l � I f I + Ig l

and the elementary properties of C (v. 1 .1 .4 , 1 . 1.5) imply that for some nonnegative constants A, B, not both zero, Af == Bg. D

[ 3.2.6 Remark. For notational consistency, when p = 1 , p' � 00 .

Furthermore, the discussions, when p = 1 , of the appropriate extensions of Holder's and Minkowski's inequalities and the criteria for equality in them, take slightly different forms. First, when f E 5,

I l f l l oo � { inf { m : I f I :::; m a.e. } if { m : I f I :::; m a.e. } -j. (/) 00 otherwIse '

and Loo � { f : f E 5, I l f l l oo < oo } . Young's inequality no longer applies when p = 1 since xp- l == 1 and x r-+ xp- l is not strictly monotonely increasing.]

If f E Ll and 9 E Loo, then

If f, g E L 1 , then

Thus , when p = 1 , both Holder 's and Minkowski's inequalities are valid. If Ig(x) 1 -j::. I l g l l oo , for some positive E and all x in a set E of positive

measure, Ig(x) 1 < I l g l loo - E. Hence

r I f gl dl-l = r + r I f gl dl-l Jx JX\E JE

< I l g l loo r I f I dl-l + ( 1 lg l loo - E) r I f I dl-l JX\E JE

< I l f l l l · l l g l l oo .

Section 3.2. The Spaces LP, 1 :s; p :s; 00 95

When p = 1, equality obtains in Holder's inequality iff

If I f + g l -:f:. I f I + I g l , for some positive E and for all x in a set E of positive measure, If(x) + g(x) 1 < I f (x) 1 + Ig(x) 1 - E. Thus, as in the calculation of the preceding paragraph, I l f + g i l l < I l f l l l + I l g l l l ' Hence, if equality obtains in Minkowski's inequality, I f (x) + g(x) 1 � I f(x) 1 + Ig(x) l . It follows (v. 1 . 1.4) that there are nonnegative functions , A, B, not both zero, such that almost everywhere, A(x)f(x) = B(x)g(x) . Almost every-

B(x) f(x) where on E¥- (fg, 0) , A(x)B(x) > 0 and A(x) = g(x) > o. When p = 1 equality obtains in Minkowski's inequality iff almost

f(x) everywhere on E¥- (fg, O) , g(x) > O.

3.2.7 THEOREM. IF p E [ 1 , (0 ) , d : (U)2 '3 (f, g) r-+ I l f - g l l p E lR SERVES AS A METRIC AND (LP, d) IS A COMPLETE METRIC SPACE. PROOF. The conventions about LP together with Minkowski's inequality and the criteria that it be an equality, imply that (LP , d) is a metric space.

When f E 5, and p E [0, (0 ) , for some e in 5 ,

(cf. the discussion following 2.4.14) . The map

is a DLS functional that generates a functional J and a corresponding space £1 . A sequence S �f {In}nEN in LP is a II l i p-Cauchy sequence iff S c £1 and S is a J-Cauchy sequence in £1 . Since £1 is J-complete, v. 2.2.32, LP is complete. D

3.2.8 Exercise. The conclusion in 3.2.7 is valid when p = 00. [Hint: Off the null set

E �f (u { x : Ih (x) 1 > I l fk l l cx,) } kEN

u ( U { x : Ifm (x) - fn(x) 1 > I l fm - fn l lcx, }) ,

{m,n}CN" {f n} nEN is a uniform Cauchy sequence.]


3.2.9 Exercise. For the map

a) 1 1 1 1 1 2 � v1J7) 2': ° and equality obtains iff 1 = 0; b) (f, g) = (g, I) ; c) for w , z in C, (wi + zg, h) = w (f, h) + z(g, h ) ; d)

is a metric for L2 ; e) (L2 , d) is a complete metric space. 3.2.10 Exercise. If S) is a vector space, and an inner product

( , ) : (S))2 '3 {x, y} r-+ (x, y) E C

is such that a)-e) of 3.2.9 obtain, for all X,y in S),

l (x, y) 1 � I Ix l 1 2 · l ly I 1 2 , I lx + y l 1 2 � I Ix l 1 2 + I ly 1 1 2 .

(3.2 . 1 1 ) (3.2 .12)

Equality obtains in (3.2 . 1 1 ) or (3.2 . 12) iff for some A, B, not both zero, Ax = By.

[Hint: For (3.2 . 1 1 ) , trivialities aside, 3.2 .9a)-d) imply that the quadratic polynomial

p(z) �f (zx + y, zx + y) is nonnegative. For (3.2 . 12) , (3.2 . 1 1 ) applies in the calculation of (x + y , x + y) .] [ 3.2.13 Note. The inequality (3.2 . 1 1 ) is Schwarz 's inequality; (3.2.12) is the triangle inequality .] When (x, y) = 0, x and y are orthogonal or perpendicular: x ..l y.

When S C S), S1- consists of all vectors y such that for each x in S, y ..l x. A subset S of S) is orthogonal (0) iff whenever X, y E S and x -j. y, then x ..l y. The set S is orthonormal (ON) iff it is orthogonal and for each x in S, I lx l l = l .

3.2.14 Exercise. a) An ON set is linearly independent. b) IfS) -j. {O} , the set oN of all nonempty orthonormal subsets of S) is a poset with respect to the order -< defined by inclusion: S1 -< S2 iff S1 C S2 . For an orthonormal set S �f {x>.} >'E!\ and an element x in S), the set <t> of all finite subsets of A is a poset with respect to the order provided by inclusion: <t> '3 ¢ -< '1jJ E <t>

Section 3.2. The Spaces LP, 1 :s; p :s; 00 97

iff ¢ C '1jJ . c) If x E S) and a>. �f (x, x>. ) , then I Ix l 1 2 2': L l a>. 1 2 ; d) The ON >'E/\

set 5 is -<-maximal iff for each x in S), I I x l 1 2 = L l a>. 1 2 iff for each x in S), >'E/\

the net n : <t> '3 ¢ r-+ n(¢) �f L a>.x>. converges to x. >'E</>

[Hint: For c) , Schwarz's inequality applies to

(L (x, x>. ) , L (x, x>. )) . >'E</> >'E</>

For d) , c) applies to prove that n(A) converges to some Y in S) (even if 5 is not maximal) and that x - y E 51- . For e) , d) and the maximality of 5 apply.] [ 3.2.15 Note. The customary name for S) is Hilbert space . The results c) resp. d) in 3.2 .14 are Bessel 's inequality resp. Parseval 's equation. A maximal orthonormal set is often called a complete orthonormal (CON) set. Hence, if {x>.} >'E/\ is a CON and

v(A) �f { !(A) if #(A) < No otherwise

(v is counting measure) , S) engenders a measure space

(A, �(A) , v) so that S) and L2 (A, v) are isometrically isomorphic.]

3.2.16 Exercise. If T �f {xn} l <n<N<No is a linearly independent subset of S), the algorithm represented by the formulre

produces an orthonormal set 5 �f {Yn} l <n<N . Furthermore, if n E N, Xn E span ( { Yk : 1 :S; k :S; n } ) , Yn E span-([xk : 1 :S; k :S; n } ) , and

span (T) = span (5) .

In particular, if T spans S), then 5 i s a CON set.


[ 3.2.17 Note. The algorithm in 3.2 .16 is known as the GramSchmidt process .]

3.2.18 Example. For L2 (A, v), the set {e), � XP}} ),E!\ is a CON set.

3.2.19 Exercise. If X is a finite set endowed with the discrete topology, p E [1 , 00] , and v is counting measure defined on !fj(X) , as sets,

C(X, q , Co (X, q , coo (X, q , and LP(X, v) , 1 � p � 00 ,

are all eX . As Banach spaces they have various norms. If f E eX , then I l f l loo = sup I f (x) 1 when f is regarded as a member of any of the first three

xEX 1

or LOO (X, v) ; when 1 � p < 00 , I l f l l p = (L I f(XW) p . If V, W is any xEX

pair of these Banach spaces , the map

id : V '3 f f-t f E W

is a norm-bicontinuous bijection. 3.2.20 Exercise. For a set X, a countable subset E of !fj(X) , and S � aR(E) , if (X, S, p,) is a-finite, then L2 (X, p,) is norm-separable. Any CON set in a norm-separable Hilbert space is finite or countable. 3.2.21 Exercise. For (X, S, p,) , if X E S, p,(X) = 1 , and f E L2 (X, p,) , then L [f(x) - L f(y) dP,(Y)] 2 dp,(x)

= L [J(XW dp,(x) - [L f(x) dp,(x)] 2 3.2.22 Exercise. a) If

X �f {O, 1 } , S �f !fj(X) , ° O.

Section 3.2. The Spaces LP, 1 :::; p :::; 00

b) If

then

X �f {o, l}n �f { y : y �f (Yl " " ' Yn ) } ' Y7 = Yi , S �f �(X), ° < p < 1 , lAy ) �f f.-ln (Yl , . . . , Yn ) = pL:=1 Yk (1 - pt-L:=1 Yk ,

f(y ) �f L�=l Yk , n

Ml �f r f(y ) df.-ln (Y) = � t r Yk df.-ln (Y) = �np = p, ix n k=)X n

M2 �f 1 [J(y)] 2 df.-ln (Y) = :2 [t 1 Y% df.-ln + L 1 YkYI df.-ln (Y)] x k=l x k#l X

1 [ ( 2 ) 2] P 2 p2 = n2 np + n - n p = ;; + p - --;; , r [J(y) - Md2 df.-ln (Y) = M2 _ M� = p(l - p) :::; � .

ix n 4n Hence

99

1 < -. - 4n (3.2.23)

3.2.24 THEOREM. (The WeierstraB Approximation Theorem) IF f E C( [O, l] , lR) AND E > 0,

THERE IS A polynomial function B SUCH THAT

sup { I f(x) - B(x) 1 : 0 :::; x :::; I } �f I l f - B l loo < E .

PROOF. There is a positive 15 such that

{ Ix - y l < r5} '* { I f(x) - f(y) 1 < � } .

If n > sup {r5-4 , 4 1 1��l oo } , then t (�) xk (l - xt-k =: 1 implies k=o

If(x) - Bn (f) (x) 1 �f If(X) - � f (�) (�) xk (l - x)n-k l = I� [f(x) - f (�) ] (�) Xk (l - x)n-k l :::; ILI *-x l <n-i 1 + ILI * -x l :;:.n-i I ·


2': 1 in the second summand of the right member above,

II (x) - Bn(f) (x) 1 � � + 2 1 1 1 1 100 � (�) xk ( l - xt-k

<: i + 2 1 1 f l l� � (�n�:) ' (�) X' (l - x)n-' E l l � "2 + 2 1 1 1 1 100 . n2 . 4n [v. (3.2.23)]

� � + 1 1 1 1 1:", . 2 2n2 If 1 1 1 1 1:", < � , then I I I - Bn(f) 1 1 < E. Thus I is uniformly approximable

2n2 2 00 by one of the Bernstein polynomials

D

3.2.25 COROLLARY. IF a < b, E > 0, AND I E C( [a , b] , lR) , THERE IS A POLYNOMIAL FUNCTION B SUCH THAT 1 1 1 - Bl loo < E. 3.2.26 COROLLARY. THE FUNCTION I I : [- 1 , 1] '3 x r-+ Ix l IS UNIFORMLY APPROXIMABLE BY POLYNOMIALS.

An alternative approach to the Holder-Minkowski-Schwarz inequalities flows from the following discussion of convex functions and their elementary properties .

3.2.27 DEFINITION. A FUNCTION ¢ : (a, b) '3 x r-+ ¢(x) E lR IS convex IFF WHENEVER ° < t < 1 AND a < p < q < b,

¢[tx + ( 1 - t)y] � t¢(p) + (1 - t)¢(q) . (3 .2.28)

3.2.29 THEOREM. IF -00 � a < b � 00, THEN ¢ IN lR(a,b) IS CONVEX IFF FOR q IN (a, b) , THE MAP (a, b) \ {q} '3 P r-+ ¢(p) - ¢(q) INCREASES MONOp - q TONELY, i.e. , IFF THE SLOPE OF THE LINE (PQ) THROUGH P � [p, ¢(p)] AND Q � [q, ¢(q)] DOES NOT DECREASE WHEN P MOVES RIGHT.

Section 3.2. The Spaces LP, 1 .s p .s 00

y-axis

a p q Figure 3.2.2.

101

p x-axis

PROOF. In English, the condition (3.2.28) says that on any subinterval of (a, b) , the graph of y = ¢(x) lies below the chord through P and Q. Thus, if ¢ is convex, as in Figure 3.2.2, then

slope(PQ) = slope(RQ) 2': slope(P'Q) , slope(PQ) .s slope(P R) = slope(P P') , (3.2 .30)

which, for the two possibilities , p < p' .s q and p < q .s p' is the burden of the assertion that the slope increases as the determining point P moves right.

Conversely, if the slope of (PQ) increases as P moves to the right , e.g. , , clef q - p' when p < P < q, and t = -- , then q - p

° < t < 1 , p' = tp + (1 - t)q,

¢(q) - ¢(p) < ¢(q) - ¢ (p' ) q - p - q - p'

,

¢[tp + (1 - t)q] .s t¢(p) + (1 - t)¢(q) .

D

3.2.31 Exercise. If ¢ : (a.b) r-+ lR is convex and a .s p < r < s < q .s b, for some constant L(p, q) , I ¢(s) - ¢(r) 1 .s L(p, q) l s - r l .

A function ¢ convex on (a, b) is Lipschitzian on every subinterval of (a, b) . Hence a convex function ¢ is absolutely continuous on every subinterval of the (open) interval that is its domain, i.e. , if E > 0,


for a positive J, if a < PI < ql < P2 < q2 < . . . < Pn < qn < b and n n

k= l k= l

[ 3.2.32 Note. In 3.2.27 the hypothesis that ¢ is defined on an open interval (a, b) is essential; e.g. , if

¢(x) �f {O �f x E [0, 1 ) , 1 1f x = 1

then ¢ is convex on [0, 1] but ¢ is not continuous when x = 1 .]

3.2.33 Exercise. a) If ¢" exists on (u, v ) , then ¢ is convex if ¢" > ° on (u, v ) . b) If ¢ is convex, then ¢" 2': ° on (u, v ) . c) exp is convex on lR.

K

3.2.34 Exercise. a) If tk 2': 0, 1 .s k .s K, L tk = 1, and {ad:= l C lR, k= l

K then min ak .s " tkak .s max ak ; b) If ¢ is convex, then, via mathe-l <k<K � l <k<K - - k= l - -

mati cal induction, ¢ (t, tkak) .s t, tk¢ (ak ) '

3.2.35 THEOREM. (Jensen's inequality) IF: a) ¢ IS CONVEX ON (a, b) ; b) (X, S , t-t) IS SUCH THAT X E S AND t-t(X) = 1 ; c) f E Ll (X, t-t) ; d) f(X ) C (a, b) , THEN ¢ (Ix f dt-t) .s Ix ¢ o f dt-t.

PROOF. If a < z < b , then 3.2.29 implies that the right-hand derivative resp. left-hand derivative i.e. ,

D+ A-( ) clef l' ¢( t) - ¢( z) D- A-( ) clef l' ¢( t) - ¢( z) 'f' Z = 1m resp. 'f' z = 1m ,

tJ·z t - z ttz t - z

exists and A �f sup D-¢(p) .s inf D+¢(q) � B. a<p<z z <q<b Hence, if A .s m .s B and a < t < b, then

¢(t) 2': ¢(z) + m(t - z) ,

i .e. , the graph of any supporting line lies below the graph of ¢. If z �f Ix f dt-t, a < x < b, and t = f(x) , then {t, z} C (a, b) and

m (f(x) - z) + ¢(z) .s ¢ [J(x)] . (3.2.36)

Section 3.3. Basic Banachology 103

Since ¢ is continuous and f is measurable, integration of both members of (3.2 .36) is permissible . Furthermore, p,(X) = 1 and z is fixed, whence 1 ¢(z) dp,(x) = ¢(z) and Jensen's inequality follows . D

[ 3.2.37 Remark. The conclusion in 3.2.34b) is an elementary instance of and a motivation for Jensen's inequality. The PROOF above is an efficient replacement for a tedious limiting argument based on approximations of f by simple functions.]

The inequality (3.2 .3) , the heart of the proof of Holder's inequality, is a consequence of the convexity of expo Indeed, if f(x)g(x) i- 0, for some

a and b in JR, exp(a) = I f(x) 1 and exp(b) = I g(x) l . Thus If (x)P + Ig(xW' , p p'

the right member of (3.2 .3) , is the convex combination

exp(ap) exp (bp') --�� + . p p'

The convexity of exp implies

( ap bP') exp(ap) exp (bp' ) If (x) 1 . Ig(x) 1 = exp -- . - = exp(a + b) .s + . p p' p p'

Jensen's inequality applied when ¢ �f exp shows that if f is measurable and JR-valued and exp( - (0 ) �f 0, then exp (1 f dp,) .s 1 exp(f) dp,.

If X �f {al , . . . , an } , Uk �f exp [J (ak ) ] , and p, (ak ) �f Ok , 1 .s k .s n, then n n n

L Ok = 1 and the preceding inequality reads II U�k .s L OkUk : k=l k=l k=l

Geometric means do not exceed arithmetic means. 1 1 In particular, if n = 2, 01 = - , and 02 = ---; , again the essence of p p

(3.2 .3) follows .

3.3. Basic Banachology

In many parts of mathematics, the study of a set 5 is carried out in part by the study of a well-chosen set M of maps of 5 into a concrete and better understood structure Y. The more the maps in M can and do respect the structure of 5, the more likely is M to reveal the nature of 5.

A map m in M respects the structure of 5 if m respects both the algebraic and topological character of 5. More specifically, if 5 is a topological vector space V, m is useful if it is a vector space homomorphism of V into


c: m(ax + by) = am(x) + bm(y) (m E [V, C] ) , whereby m is respectful of the algebraic structure; and m is continuous : if n : A '3 A r-+ n(A) E 5 is a net and n(A) converges to x, then m [n(A) ] converges to m (x) , whereby m is respectful of the topological structure. Such an m is a continuous linear functional.

The set of all continuous linear functionals on V is its dual space or dual denoted V', [V, ej c or simply [V] c . When the elements of V are denoted v, . . . , the elements of V' are denoted v' , . . . and when v E V and v' E V', the number v' (v) is written (v , v' ) . Thereby a v in V reveals itself as an element of (V')' �f V". In analogy with the situation in 5) , when clef { } 5' clef { ' } , 5

= VA AE/\ C V resp. = VA AE/\ C V ,

1- clef { ' , , ( ' ) } 5 = v : V E V , VA , V = 0 , , clef { ( ' ) _ } 51- = v : v E V, V' VA = 0 .

3.3.1 DEFINITION . THE TOPOLOGICAL VECTOR SPACES V, W CONSTITUTE A dual pair {V, W} IFF: a) V C W' AND W C V'; b) FOR v resp. v' FIXED, (v , v') = 0 IFF v' = 0' resp. v = O.

[ 3.3.2 Note. The set of not necessarily continuous linear maps of V into C is denoted V* by some. In extensive treatments of the subject of topological vector spaces, e.g, [Kot, Sch] , the distinction between [V, ej , and [V, ej c (= V') is explored at some length. By definition, V' C V* . The Hausdorff Maximality Principle implies that a vector space V contains a maximal linearly independent subset H �f {xAhE/\' usually called a Hamel basis . When the topological structure of V is sufficiently rich, e.g., when V is an infinite-dimensional Banach space such as L1 ( [0 , 1] , A) , a Hamel basis for V leads to the conclusion V' ¥V*, [Ge3, GeO] .

The results in Section 3.2 imply that if p E [1 , (0) and V = LP, then Lpi C V'. Later arguments (v. Section 4.3) show that in mildly restricted circumstances: a) if p E [1 , (0) , then (LP)' = LP' ; b) if X is a locally compact Hausdorff space, corresponding to each continuous linear functional F operating on one of Co(X, q or Coo (X, q , there is a complex measure space (X, 5 (3, p,) such

that for each f in Co (X, q or Coo (X, q , F(f) = Ix f dp,.]


3.3.3 THEOREM. IF (B, I I I I ) AND (F, I I I I ) ARE BANACH SPACES, A T IN [B, F] IS CONTINUOUS IFF EITHER: a) T IS CONTINUOUS AT 0; OR b) FOR SOME K IN [0, (0) AND ALL x IN B, I IT(x) 1 1 .s K l lx l l . PROOF. Since lim xn = x iff lim (xn - x) = 0, the linearity of T implies n-+CXJ n-+CXJ the truth of a) .

For b) note that if some K as described exists, T is continuous at o . Conversely, if no such K exists, for each n in N, there is an Xn such that l iT (xn ) 1 1 > n I lxn l l or, equivalently, l iT CI:: I I ) II �f l iT (Yn) II > n. How-

ever, I IYn l 1 == 1 , whence }�� = 0, whereas l iT (�) I I > yn, in contradiction of the continuity of T at O. D

[ 3.3.4 Note. In the context above, the norm of T is

I IT I I �f inf { K : I IT(x) II .s Kl l xl l } . ]

3.3.5 Exercise. With respect to the norm defined in 3.3.4, the set [B, F]c of continuous elements of [B, F] is a Banach space. 3.3.6 Exercise. For T in [B, F]c ,

I IT I I = sup { I IT(x) 1 1 . x "l o } II x i i . = sup { I IT(x) 1 1 : O .s II x i i .s I } = sup { I IT(x) 1 1 : I lx l l = I } { I (T(x) , x') I I ' } = sup II x i i . I l x' l l : I l x I · I lx I I "I ° .

The results that follow are at the heart of Banachology. The first is formulated in terms of a seminorm, i.e. , for a vector space V, a map

p : V '3 x r-+ p(x) E [0 , (0 ) such that p(x + y) .s p(x) + p(y) and, for z in C, p(zx) = I z lp(x) . 3.3.7 THEOREM. (Hahn-Banach) IF p IS A SEMINORM ON A VECTOR SPACE V, W IS A subspace OF V, m E [W, q , AND FOR ALL w IN W,

1m (w) 1 .s p (w) , THERE IS IN [V, q AN m SUCH THAT FOR ALL x IN V,

Im(x) 1 .s p(x)


AND m lw = m: m IS AN EXTENSION OF m. PROOF. The argument is carried out in three steps: a) It is assumed that m(W) c JR and for some y in V, W ¥ U = { w + zy : w E W, z E JR } . b) One of the equivalents of the Hausdorff Maximality Principle is used to remove the restriction on U. c) The restrictions m(W) c JR and z E JR are removed.

a) For all w in W, some a in JR, and all z in JR, the formula

m(w + zy) �f m(w) + za

defines a linear functional on U and m lw = m. The condition

Im(u) 1 :::; p(u)

for all u in U places a demand on a, since Im(w + zy) 1 :::; p(w + zy) for all w in W iff for all nonzero z and all w in W,

Im(-zw + zy) 1 :::; p(-zw + zy)

iff (after division by I z l ) Im(w) - a l :::; p(w - y) = p(y - w) iff

m(w) - p(y - w) :::; a :::; m(w) + p(y - w) . (3.3.8)

(A similar argument appears in the PROOF of 3.2.35 where the existence of an m in [ sup D� ¢(p) , inf D+ ¢( q)] is used. )

a<p<z z <q<b For any u, v in W,

m(u) -m(v) = m(u-v) :::; p(u-v) = p(u-y+y-v) :::; p(u-y) +p(v-y), m(u) - p(u - y) :::; m(v) + p(y - v) ,

sup [m(u) - p(u - y)] :::; inf [m(v) + p(y - v)] . uEW yEW

Since m lw = m, at least one a satisfying (3.3.8) exists. b) The set S of proper supers paces of W to which m can be extended

correctly is nonempty since S includes U. With respect to consistent inclusion as order, i.e. , U1 --< U2 iff U1 c U2 and the extension of m to U2 coincides on U1 with the extension of m to U1 , S is a poset. The Hausdorff Maximality Principle applies and provides a maximal extension M of m. If the domain of M is not V, the discussion in a) implies a contradiction of the maximality of M: M is defined on all V.

c) If m(W) c C and r(w) � � [m(w)] , on W,

Ir(w) 1 :::; Im(w) 1 :::; p(w) .


The arguments in a) and b) apply to r since a vector space over C is automatically a vector space over JR.. If R is the extension of r to V, then

M : V '3 x f-t R(x) - iR(ix)

maps V into C and M is an extension of iii . If x E V and eiQ = sgn [iVf(x) ] , then eiQ M(x) = IM(x) 1 {= � [eiQ M(x)] } and

IM(x) 1 = M (eiQx) = R (eiQx) :s; p (eiQx) = p(x) . D

3.3.9 COROLLARY. IF B IS A BANACH SPACE AND 0 i- x E B, THERE IS IN B' AN x' SUCH THAT I lx' l l = 1 AND (x, x') i- O.

PROOF. The set W �f { zx : z E C } is a subspace of B. The formula

iii : W '3 zx f-t iii(zx) �f z l lx l l E C

defines a linear functional on W and iii(x) = I lx l l i- o. With II I I serving as p in 3.3.7, it follows that for some x' in B' , x' lw = iii and for each y in B, Ix'(y) 1 :s; I l y l l · The result in 3.3.6 implies I lx' l l :s; 1 , and since (x, x' ) = I l x l l , it follows that I l x' l l = 1 . D

3.3. 10 COROLLARY. IF W IS A CLOSED SUBSPACE OF A BANACH SPACE B AND x tJ. W, FOR SOME x' IN B' ,

(x, x' ) = 1 , x' (W) = {O} ,

i.e. , x' E Wi- ; IF d �f inf { I l x - wl l : w E W } , THEN I lx' l l :s; � . PROOF. The subspace Y �f { zx + w : z E C, w E W } is closed. The formula iii : Y '3 zx + w f-t iii(zx + w) �f z defines a linear functional on Y and iii(W) = {O} . If z i- 0, I l zx + wl l = I z l · l lx + 7 1 1 2': I z l d, whence

liii(zx + w) 1 = I z l :s; � I l zx + wl l · (3.3 . 1 1 )

Hence iii E [W, Clc , and the Hahn-Banach Theorem (3.3.7) implies for some x' in B', X' l y = iii, (x, x') = 1 , x' E Wi- . Moreover (3.3 . 1 1 ) implies

I lx' l l :s; � . D


. , I ' 1 3.3. 12 ExercIse. For x as in 3.3. 10, I x II = "d : [Hint: For some sequence {wn}nEN contained in W, I lwn - xi i + d and 1m (wn - x) 1 = 1 :::; I l x' l l · l lxn - w l l ·]

3.3.13 Exercise. If {vn}nEN is a linearly independent set in a Banach space the statements:

clef Xl = VI , for some x� , (x 1 , x� ) = 1 ,

n- l Xn � Vn - L (Vk , X�_ l ) Xk , k=l

for some x�, x� E span [(Xl ' . . . ' xn_d] 1- , and (xn , x�) = 1 ,

engender an analog of the Gram-Schmidt process and lead to a biorthogonal pair ({ xn} nEN ' {x�J nEN) such that for m, n in N,

if m = n clef s: (K k ' f . ) . = Urnn ronec er s unction , otherwIse

and for which span ({Vn}nEN) = span ({xn}nEN) . 3.3.14 Exercise. A proper closed subspace M of a Banach space B is nowhere dense in B.

COROLLARY 3.3.10 implies that for any Banach space B, B' is not only nonempty but is equipped with a ready supply of separating elements that distinguish any nonzero x from ° and hence any two elements of B from one another: if x i- y, for some x' in B', (x - y, x') i- 0, i .e. , (x, x' ) i- (y , x' ) . 3.3.15 Exercise. With respect to I I I I as defined in 3.3.4, B' is a Banach space.

Although topological completeness plays no role in the Hahn-Banach Theorem, topological completeness is an essential ingredient in the next results.

3.3.16 THEOREM. (The Open Mapping Theorem) IF B AND F ARE BANACH SPACES , T E [B, F]c , AND T(B) = F, THEN T IS OPEN: A CONTINUOUS LINEAR SURJECTION ( A CONTINUOUS epimorphism) OF ONE BANACH SPACE ONTO ANOTHER IS OPEN . PROOF. To show T is open, it suffices to show that for some positive p, T [B(O, It] contains some B(O, pt .

Since B = U B(O, nt , it follows that F = U T [B(O, nt] · Be-nEN nEN

cause F is complete, F is of the second category, i.e. , F is not the union


of count ably many nowhere dense sets. Thus some T [B (0, not] is not nowhere dense, i .e. , T [B (0, not] is somewhere dense: for some positive R and some y, T [B (0, no )O ] contains B(y, Rt �f W. Since for some x, T(x) = y, if I lx l l = D, then

T [B (O, no + D)O] :::) T [-x + B (O, no )O] :::) B(O, Rt ,

T [B(O, 1) °] :::) B (0, R ) ° �f B(O, rt. no + D

(The technique just used may be described as translation/scaling . ) If E E (0, 1 ) and Z E B(O, rt, for some Xl in B(O, I t ,

(whence Z - Z l E B(O, Ert ) . For some X2 in B(O, Et,

(whence Z - Z l - Z2 E B (0 , E2r) 0 ) . Induction yields sequences {xkhEN {zkhEN such that for k in N,

I lxk l l < Ek- l (whence Xk E B (O, Ek- I ) O) ,

( ) clef B (0 k- I ) ° T Xk = Zk E , E r ,

l iz - t. Zi I I < Ekr.

Hence x( �f f Xk E B ( 0, 1 � E) and, owing to the continuity of T, k=l

T (x( ) = z. Thus T [B (0, 1 � E) 0] :::) B(O, rt. Since T is linear,

T [B(O, It] :::) B[O, (1 - E)r] O �f B(O, p) . D

3.3.17 THEOREM. FOR BANACH SPACES B AND F, IF T E [B, F]c , T- I EXISTS, AND T(B) = F, THEN T-I E [F, BJ c .

A continuous linear bijection between Banach spaces is bicontinuous.

PROOF. The graph 9 �f { {x, T(x) } : x E B } of T is a subset of B x F. Normed according to the formula I I {x, T(x) } 1 1 � I lx l l + I IT(x) l l , B x F is


a Banach space and 9 is a closed subspace, hence is itself a Banach space. Moreover, for f : 9 '3 {x, T(x) } f-t T(x) E F, f-1 exists . The Open Mapping Theorem (3.3.16) implies f is open, whence f- l is continuous. Since 71"1 : 9 '3 {x, T(x) } f-t x E B is (automatically) continuous (v. 1 .2. 15 and 1.7.19) , and since 71"1 o f-1 = T- 1 , T-1 is continuous. D

3.3.18 COROLLARY. (The Closed Graph Theorem) IF B AND F ARE BANACH SPACES , T E [B, F] , THEN T E [B, F] c IFF THE GRAPH 9 (T) � 9 IS CLOSED IN B X F NORMED AS IN THE PRECEDING PROOF.

PROOF. If 9 is closed, it is a Banach space. The map

f : B '3 x f-t [x, T(x) ] E 9

is in [B, 9] , f-1 exists, and f- 1 E [9, B] . If

then f-1 [xn, T (xn ) ] = Xn and, since lim Xn = x, f- l is continuous. The n�= Open Mapping Theorem implies f-1 is open, whence f is continuous . Since

71"2 : 9 '3 [x, T(x) ] f-t T(x) E :F

is also automatically continuous, T (= 71"2 0 f) is continuous. Conversely, if T is continuous, and [xn , T (xn ) ] converges to some (x, y)

in B x F, then Xn converges to x and the continuity of T implies T (xn ) converges to T(x) : y = T(x) , i .e., (x, y) E 9 , 9 is closed. D

The context for 3.3.18 is the theory of Banach spaces. In a more general context of topology there is the following analogous result.

3.3.19 Exercise. If X and Y are compact topological spaces, f E yX ,

the graph of f, 9(1) � { (x, y) : y = f(x) } is closed iff f is continuous. That a Banach space is a set of the second category is essential in the

PROOF of

3.3.20 THEOREM. (Banach-Steinhaus) IF B AND F ARE BANACH SPACES,

AND FOR EACH x IN B, sup I IT), (x) 1 1 < 00, FOR SOME POSITIVE M, ),E/\

sup l iT), II :::; M. ),E/\

Section 3.4. Weak Topologies 1 1 1

PROOF. If, for n in N, En �f { X sup I IT), (x) 1 1 :::; n } , then B = U En. ),EA nEN

Thus some En is somewhere dense and, by virtue of the translation/scaling device used in the PROOF of the Open Mapping Theorem (3.3. 16) , for some positive r, El is dense in B(O, rt . Hence, for all x in El ,

sup I IT), (x) 1 1 :::; 1 . ),EA

If I l z l l :::; r, some sequence {xn}nEN contained in El converges to z. For each >',

I IT), (z ) 1 1 :::; l iT), (xn ) 1 1 + I IT)' I I · l l z - xn l l · (3.3.21 )

For large n, the second term in the right member of (3.3.21 ) is small, whereas the first term in the right member does not exceed 1 : l iT), I I :::; � . r When the translation/scaling is reversed, the required assertion follows.

D

3.3.22 DEFINITION. A SEQUENCE S �f {Xn}nEN IN A NORMED (VECTOR) 00

SPACE V IS summable IFF L xn E V . THE SEQUENCE S IS absolutely n=l

00 summable IFF L I l xn l l < 00.

n=l 3.3.23 Exercise. a) A normed vector space V is complete iff every absolutely summable sequence is summable. b) The result in a) offers another proof that LP is a complete metric space.

[Hint: If: If {xn} nEN is a Cauchy sequence, for some subsequence {Xnk } kEN' {Xnk+1 - Xnk } kEN is absolutely summable. A modification of the argument in 3.2.7 applies.

N

Only if: The partial sums Sn �f L Xn form a Cauchy sequence.] n=l

3.4. Weak Top ologies

The results in Section 3.3 deal with the uniform or norm-induced topology for the set [B, F]c of continuous linear operators between the Banach spaces B and F. For some important invpstigations , other topologies are more useful.

For a Banach space B, the sequence

B, B' , (B') ' �f B", . . .


is meaningful. For any fixed x in B, (x, x') is a continuous function on B' and thus x may be regarded as an element of B". 3.4.1 Exercise. If B is a Banach space, the map � that identifies each x in B with its correspondent in B" is an injection, and for each x in B,

1 1 � (x) 1 1 = I lx l l ·

3.4.2 Exercise. If B, F are Banach spaces , T E [B, F] and for each x in B, I IT(x) 1 1 = I l x l l (T is an isometry) , then T E [B, F] c and T(B) is a closed subspace of F. (Hence � (B) is a closed subspace of B".)

Owing to the last two results , whenever convenience is served, no distinction is drawn between B and �(B) .

3.4.3 DEFINITION . THE BANACH SPACE B IS reflexive IFF �(B) = B". 3.4.4 Exercise. The Baill\ch space B is reflexive iff B' is reflexive. (hence, ·ff B" B'" fl . ) I , , . . . are re eXlve .

For an infinit�dimensional Banach space B and its dual B' there are two important topologies different from those induced by their norms.

3.4.5 DEFINITION . FOR THE DUAL PAIR {B, B'} OF BANACH SPACES, a (B, B') resp. a (B', B) IS THE WEAKEST TOPOLOGY SUCH THAT EVERY x' resp. x IS CONTINUOUS ON B resp. B' . THESE TOPOLOGIES ARE THE weak resp. weak! TOPOLOGIES FOR B resp. B' . THE NOTATIONS BW resp. (B' )W'

ARE USED TO SIGNIFY B resp. B' IN ITS WEAK resp. WEAK ' TOPOLOGY. 3.4.6 Exercise. a) For a Banach space B the set

N (0; x� , . . . , x� ; E) � { x : x: E B', E > 0, I (x, x; ) I < E, 1 :::; i :::; n } ,

is a convex a (B, B')-neighborhood of O. Dually,

is a convex a (B', B)-neighborhood of 0'. Furthermore, each such neighborhood is circled, i.e., if ), E C and 1 )' 1 :::; 1 , then )'N c N.

b) The set NW resp. NW' of all sucR a (B, B')-neighborhoods resp. a (B' , B)-neighborhoods is a base of neighborhoods at 0 resp. 0'.

c) The sets x� , . . . , x� and Xl , . . . , Xn may be chosen to be linearly independent without disturbing the conclusions in a) .

d) With respect to these topologies B and B' are LCTVSs. 3.4.7 Exercise. For a Banach space B, the weak resp. weak' topology for B resp. B' is weaker than the norm-induced topology. The weak resp. weak' topology is the same as the norm-induced topology iff dim (B) E N.

Section 3.4. Weak Topologies 113

3.4;8 LEMMA. IF B IS A BANACH SPACE, FOR THE TOPOLOGIES (J" (B, B') AND (J" (B', B), {B, B'} IS A DUAL PAIR AND EACH MEMBER OF THE DUAL PAIR IS THE DUAL OF THE OTHER.

PROOF. Since B and � (B) may be regarded as indistinguishable, a) in 3.3.1 is satisfied. If x' E B' and (x, x' ) == 0, then x' = 0' by definition. If x i- 0 the Hahn-Banach Theorem (3.3.10) implies that for some x', (x, x' ) ;f:. 0, whence b) in 3.3.1 is also satisfied.

If m E (BW )' , since (J" (B, B') is weaker than the norm-induced topology, m is norm-continuous, whence (BW)' c B'. On the other hand, if x' E B' and E > 0. for any x in N (0; x' ; E) , I (x, x' ) 1 < E, whence

If m E ((B')W' ) ' there is a weak' neighborhood

such that {x I , . . . , xn} is linearly independent and if y' E N, then

1m (y') 1 < 1 .

For any y', if a = sup I (Xk , y' ) I , then rSy' E N, l �k�n 20' ( rSy,) whence m 20' < 1 ,

i.e., m (y') < � sup I (Xk , y') I . In particular, if y' E [span (xl , . . . , xn )] i- , u l �k�n

then m (y') = 0. For the biorthogonal pair (cf. 3.3.13) { {Yk} 19�n ' {YDI�k�n} asso

ciated with {Xl , . . . , xn} , if z' E B', then

z'

= � (Yk , Z' ) Y� + (z' - � (Yk ' Z' ) Y�) � u' + v'

,

m (z') = m (u') + m (v') .

Since v' E [span (Xl , . . . , Xn ) ]i- ,

n Consequently, m may be identified with L O'kYk ·

k=l D

In a topological vector space V, a set S containing {O} is absorbent iff for each v in V there is a nonzero t such that tv E S.


3.4.9 Exercise. For a topological vector space V and a neighborhood N of 0:

a) N i s absorbent; if a -j. 0, then aN i s a neighborhood of 0; U AN 1>' 1< 1

is circled; for some positive E, if I r l < E then rN C N; at 0 there is a base of circled neighborhoods;

b) A(x, N) �f { a : 0' 2': 0, x E aN } -j. 0; if N is circled and

PN : V '3 x r-+ inf { a : 0' 2': 0, x E aN } ,

then: bI) { {a E A(x, N)} 1\ {;3 > a}} '* {;3 E A(x, N) } ; b2) for some N, {x -j. O} '* {PN(X) -j. O} ; b3) PN(O) = 0; b4) for t in C, PN (tX) = I t lpN (X) ; b5) if N is convex, then PN(X + y) :::; PN (X) + PN(Y) ;

c) if PN is a function for which bI )-b5) obtains, then { x : PN (X) .} >. E/\ of functions conforming to b I )� b5) , the set N �f { N>. : N >. � { x : P>. (x) < 1 } A E A } is a base of neighborhoods at 0, and for each v in V \ {O}, there is a A such that v tJ- N>. .

[Hint: b5) : If E > 0, for some positive a and ;3, {� , �} e N,

0' - E < PN(X) :::; a and ;3 - E :::; PN(Y) :::; ;3. Furthermore,

X + Y = _a_ � + _;3_ � E N.

0' + ;3 0' + ;3 0' 0' + ;3 ;3

Hence PN(X + y) ::; 0' + ;3 < PN(X) + PN (Y) + 2E. ]

[ 3.4.10 Note. The function PN is the Minkowski functional associated to the neighborhood N [Kot, Sch] . Owing to c)�d) , PN is a seminorm. If A(x, N) = 0, by definition, PN (X) = 00. If, for each x and each N, A(x, N) -j. 0, PN is a norm.]

3.4 .11 LEMMA. IF B IS A BANACH SPACE, THEN � (B) IS a (B", B')-DENSE IN B". PROOF. Otherwise for some weak' neighborhood

N � N (0") � { Y" : l (x� , Y") 1 < E, 1 ::; k :::; n }

Section 3.4. Weak Topologies

and some x" in B", (x" + N) n � (B) = 0. For w" in

L �f span [x" + �(B)] ,

h L " clef " c ( ) . l' f L ' rr t e map m : '3 w = ax + " x r-+ a IS a mear map 0 mto \L- .

115

Since (x" + N) n �(B) = 0, PN [x" + �(x)] 2': 1 . Since N is circled, and �(B) is a vector space, if x E B and a -j. 0, then

i.e., Im (w") 1 :::; PN (w") , an inequality that remains valid when a = 0. The Hahn-Banach Theorem (3.3.7) implies that there is a linear func

tional m : B" '3 u" r-+ m (u" ) E C such that

Furthermore, if 15 > ° and u" - v" E r5N, then PN (u" - v") :::; 15, whence 1m (u" - v") 1 :::; 15. In short, m is a (B", B')-continuous . However, 3.4.8 implies that for some z' in B', m (u") = (z'

, u" ) . Since m[�(B)] = {a} , it follows that z'

= 0', whence m is the zero functional. However, m (y") = 1 , a contradiction. D

3.4.12 LEMMA. (Alaoglu) IF B IS A BANACH SPACE, THEN B (0', 1 ) IS a (B', B)-COMPACT.

PROOF. For each x in B, { (x, x') : x' E B (0', I ) } C [0, II x i i ] � Ix . Tychonov's Theorem implies K �f XXEBlx is compact in the product topology T. The map (J : B (0', 1 ) '3 x' r-+ (J (x') �f {(x, x' ) }xEB(O, I ) E K is , by virtue of the Hahn-Banach Theorem, injective and a (B', B) - T continuous , whence on (J [B (0', 1 )] �f Y, (J- l is T - a (B', B) continuous . For x, y in B, and a in C, the maps

�x,y : K '3 {aX}xEB r-+ ax+y - ax - ay E C, 1]n,x : C x K '3 (a, {aX}xEB) r-+ O'ax - anx E C,

are continuous . Hence, for each map, the inverse image kx,y resp. kn,x of {a} is closed. Thus, (J [B (0', 1 ) ] = Y = (n kx,y) n (n kn,x) is a

X,Y nIx closed, hence compact set. It follows that B (0', 1 ) is a (B', B)-compact.

D

3.4.13 THEOREM. THE BANACH SPACE B IS REFLEXIVE IFF B(O, l ) IS WEAKLY COMPACT.


PROOF. If B is reflexive, then � (B) = B", � IB(O, 1 ) ] = B (0" , 1 ) , and the topology inherited by � [B(O, 1) ] from the weak' topology of B" is the same as the weak topology of B(O, 1 ) . By virtue of 3.4.12 , B(O, l ) is weakly compact .

Conversely, if B(O, 1) is weakly compact, the a (B, B') - a (B", B')continuity of � implies � [B(O, 1)] is a (B", B')-compact . However, 3.4 .11 implies � [B(O, 1 ) ] i s a (B" , B')-dense in B (0" , 1 ) , whence

� [B(O, 1 ) ] = B (0", 1 ) ,

which implies �(B) = B". D

3.4.14 DEFINITION. FOR BANACH SPACES B AND F AND T IN [B, Fl c THE ADJOINT T' IS THE UNIQUE ELEMENT OF [F' , B']e SUCH THAT FOR EACH (x, y' ) IN B X F' , [x, T' (y' ) ] = [T (x) , y'] .

[ 3.4.15 Remark. The notation T' for the adjoint is consistent with the notation V' for the dual space. Some writers use V* instead of V'; correspondingly they use T* instead of T'.]

3.4.16 Exercise. a) The statement in 3.4.14 is meaningful, i.e. , T' exists and is unique. b) I IT' I I = I IT I I . c) If {a , b} C C and {S, T} C [B, F] e , then (as + bT)' = as' + bT' .

[Hint: a) The Closed Graph Theorem (3.3.18) applies. b) 3.3.6 applies .]

3.5. Banach Algebras

Gelfand [Gelf] introduced the notion of a normed ring, known today as a Banach algebra. It combines the concepts of Banachology and algebra to form a discipline with many useful developments. Only the outlines of the subject are treated below. Details are available in [Ber, HeR, Loo, Nai, Ri) .

Some Banach spaces , e.g. , function algebras such as Co (lR., C) , form the context for introducing not only addition and scalar multiplication of their elements but also a kind of addition-distributive product of elements. The basic aspects of this development are treated below.

3.5.1 DEFINITION. A BANACH SPACE A THAT IS ALSO A C-algebra IS A Banach algebra IFF FOR a AND b IN A AND z IN C:

I l ab l l :::; I l al l l l b l l ; z(ab) = (za)b; I l zal l = I z i l l al l ·

Section 3.5. Banach Algebras 117

3.5.2 Example. When X i s a locally compact Hausdorff space, the Banach space Co (X, q �f A, normed by I I 1 100 , is a commutative Banach algebra with respect to pointwise multiplication of its elements; A has a multiplicative identity e iff X is compact, in which case e = l .

3.5.3 Example. For a Banach space B, the set [B] c cl�f A of continuous endomorphisms (of B), normed according to the discussion in 3.3.4, is a Banach algebra with respect composition of its elements; A is commutative iff dim (B) :::; 1; the identity endomorphism id is always the identity for A.

3.5.4 Exercise. If the Banach algebra A contains a multiplicative identity e such that ea == ae == a: a) e is the only identity; b) renormed according to I l x l l ' �f sup I l axl 1 1

1 1, A is again a Banach algebra and I l e l l ' = 1; c) for some

#0 l a positive K and all x, K l lx l l :::; I lx l l ' :::; I lx l l · Thus II I I and I I I I ' are equivalent norms :

If A contains an identity e, I l e l l may be taken as 1 .

3.5.5 DEFINITION. WHEN A BANACH ALGEBRA A CONTAINS AN IDENTITY clef e, Ae = A; WHEN A CONTAINS NO IDENTITY, e IS A SYMBOL SATISFY-clef { } ING e tJ. A AND Ae = ze + x : z E C, x E A . WHEN A CONTAINS NO IDENTITY AND {Zie + Xi}i=1 ,2 C Ae ,

Z l e + Xl + Z2e + X2 �f (Zl + Z2 ) e + (Xl + X2 ) , (Zl e + xd (Z2e + X2 ) � Zl Z2e + ZlX2 + Z2Xl + XIX2 ,

AND Ae IS NORMED ACCORDING TO I l ze + X i i � I z i + I IX I I . If A = Ae and ab = e, a is a left inverse of b and b is a right inverse

of a. 3.5.6 Exercise. For a Banach algebra A : a) Ae is a Banach algebra; b) the map A '3 x r-+ Oe + x E Ae is an isometry.

3.5.7 Exercise. If A = Ae and u and v are left and right inverses of x, h clef - 1 d 1 f . ( . h . ) f . - 1 t en u = v = x an every e t Inverse ng t mverse 0 x IS X .

3.5.8 Example. When v is counting measure, the classical Hilbert space (v. Section 3.6) is L2 (N, v) �f £2 consists of all vectors a �f (al , a2 , . . . ) 00 of complex numbers such that L l an l 2 < 00. The set [p2L of continuous

n= l


endomorphisms of A is , with respect to composition of endomorphisms as product , a noncommutative Banach algebra. The maps

T : £2 '3 (al ' a2 , . . . ) f-t (a2 ' a3 , . . . ) , S : £2 '3 (aI , a2 , . . . ) f-t (0 , aI , a2 ' . . . ) .

are continuous endomorphisms. Furthermore, T S = id but ST i- id . Thus T[S + (id - ST)] = id : both S and S + (id - ST) are different right inverses of T: absent commutativity, right inverses need not be unique. Since S'T' = id ' = id and T'S' i- id a similar argument shows left inverses need not be unique.

In Ae , the identity (e - x) ( e - y) = e - (x + y - xy) introduces the expression x + y - xy. If e - x has a right inverse, it may be written as e - y, and (e - x) ( e - y) = e or x + y - xy = 0: x + y - xy is meaningful even when A contains no identity.

3.5.9 DEFINITION. FOR x AND Y ELEMENTS OF A BANACH ALGEBRA A, x 0 Y �f X + Y - xy. WHEN x 0 Y = 0 , Y IS A right adverse OF x (AND x IS A left adverse OF y) . 3.5.10 THEOREM. THE BINARY OPERATION 0 IS ASSOCIATIVE. IF U AND clef ° d V ARE LEFT AND RIGHT ADVERSES OF X, THEN U = V = X , THE a verse OF x AND xxo = xOx. PROOF. The associativity of 0 follows by direct calculation.

If U 0 x = x 0 V = 0 , then

u = u 0 0 = u 0 (x 0 v) = (u 0 x) 0 v = 0 0 v = v � XO and u = xu - x = ux - x. Hence xOx = xux - x2 = xxo . D

3.5.11 Exercise. a) The adverse XO exists iff (e - X)- l exists in Ae. b) 00

If I lx l l < 1 , then XO exists and XO = - L xn . c) In Ae, n=l

(x o y) (e - x) = (e - x) (y o x) ; d) If XO exists, for any z,

(z - XC ) (e - x) = z 0 x and (e - x) (z - XC ) = x 0 z.

e) If XO and yO exist , then x 0 y i s advertible and (x 0 yt = yO 0 xo . f) In Ae , if l i e - xi i < 1 , then X-I exists. g) In Ae, if X-I exists, for some positive r, y- l exists if I ly - xi i < r .

In Ae the set H of invertible elements is nonempty and open.

Section 3.5. Banach Algebras

00 [Hint: f) The series e + 2:)e - x)n converges and

n=1 [e - (e - x)] (e + �(e - x)n) = e. 1

g) If I I v i i < I I x- I I I ' f) implies e + X- IV i s invertible.]

[ 3.5. 12 Note. The result g) has a counterpart for adverses, v. 3.5. 14.]

1 19

3.5.13 Exercise. a) In Ae the set of H of invertible elements is : a group relative to the operation of multiplication in A. b) The map

H '3 x f--t X- I E H is a bicontinuous bijection, i.e. , an auteomorphism.

3.5.14 DEFINITION. FOR A BANACH ALGEBRA A, THE SET adv (A) CONSISTS OF THE ADVERTIBLE ELEMENTS , i .e . , THOSE FOR WHICH THERE IS A (UNIQUE) LEFT AND RIGHT ADVERSE.

3.5.15 LEMMA. FOR A BANACH ALGEBRA A, adv (A) IS AN OPEN SUBSET (cf. 3.5. 11g) ) AND ° : adv (A) '3 g f--t gO IS AN AUTEOMORPHISM.

1 PROOF. If x E adv (A) and I ly - xi i < 1 + I lxo l l ' then

XO 0 Y = (y - x) - XO (y - x) , y o XO = (y - x) - (y - x)XO ,

I lxo 0 yl l � I ly - xi i (1 + IlxO I I ) < 1, I ly 0 x0 1 1 � I ly - xi i ( 1 + IlxO I I ) < 1,

whence XO 0 Y and y 0 XO are advertible. If w is the adverse of XO 0 y, then w 0 XO is a left adverse of y. If z is the adverse of y 0 xO , then XO 0 z is a right adverse of y. Thus y is advertible: adv (A) is open.

If x E adv (A) and x + h E adv (A) , then

I l h - hxo l l � I l hl l (1 + IlxO I I ) , and if I l hl l is small, then u �f h - hxo E adv (A). Furthermore, 3.5. 11 implies

(x + h) o xo = u, (x + ht = XO 0 uO ,

(x + h) ° - XO = UO - xOuo ,


Since I l ul l � I l hl l ( l + I lxl ! ) , when I l hl l is small enough, I lul l < 1. Thus

I l hl l ( 1 + I lxO I I ) 2': I l ul l = I l uo - uuo l l 2': I luo l l - I lul l I l uo l l , ° < l I ul l < I l hl l ( 1 + I lxO I ! ) I lu I I - 1 - Il ul l - 1 - Il hl l (1 + I l xO I I ) '

whence the map ° : adv (A) f-t adv (A) is continuous . The symmetry of the definition of x 0 y implies that (uot = u and thus adv (At = adv (A) .

D

3.5.16 THEOREM. IF x IS AN ELEMENT OF A BANACH ALGEBRA, THEN 1

lim I lxn l l n EXISTS. IT IS DENOTED sr (x) AND THERE OBTAINS THE n--+= 1

EQUATION sr (x) = inf I l xn l l n . FURTHERMORE: a) 0 � sr (x) � I lxl l ; b) nEN FOR z IN C, sr (zx) = I z lST (x) ; c) sr (xy) = sr (yx) , AND FOR n IN N, sr (xn ) = [sr (x)t; d) IF xy = yx, THEN sr (xy) � sr (x)sr (y) .

1 PROOF. If E > 0, for some m in N, I lxm l l m � sr (x) + E. If n E N, for some p, q in N, n = pm + q, 0 � q < m, whence

pm As n -+ 00, - -+ 1 , whence n _ 1 1 1 lim I lxn l l n � sr (x) = inf I lxn l l n � lim I lxn l l n , n--+CXJ nEN n--+CXJ

i .e. , lim I lxl l -i;.- exists and is sr (x) . n--+= Items a)-d) follow by direct calculation. D

3.5.17 DEFINITION . FOR x IN A BANACH ALGEBRA A, THE spectrum of x IS sp(x) �f { z : (x - ze) - I does not exist in Ae } .\ 3.5.18 Exercise. a) If A ¥Ae and x E A, then

sp(x) = {O} U { z : z i- 0, (z- lx) O does not eXist } .

b) If A = Ae, x E A, and z i- 0, then z E sp(x) iff (�) ° does not exist; o E sp(x) iff X- I does not exist, i.e., iff x is singular.

[Hint: If z i- 0: a) xy = e iff z- Ix · zy = e; b)


3.5.19 THEOREM. IF x IS AN ELEMENT OF A BANACH ALGEBRA, FOR SOME Z IN Sp(X) , sr (x) :::; I z i .

[ 3.5.20 Remark. Hence sp(x) i- 0.]

PROOF. (Rickart [RiD If 0 tJ. sp(x) , then A = Ae and X- I exists. Hence 3.5. 16d) implies 1 = sr (e) :::; sr (x) sr (X- I ) . Thus, if sr (x) = 0, then X- I does not exist, i .e. , 0 E sp(x) : the result is established if sr (x) = o.

If sr (x) �f p > 0, the idea of the argument is to show by contradiction that sp(x) ct D(O, pt . Indeed, if sp(x) C D(O, pt, then

{ Iw l � p} =? {w tJ. sp(x) } ,

i .e. , (w- lx) O exists. From 3.5.15 it follows that

clef ( 1 ) ° W r-+ f(w ) = w- x

is a continuous map from S �f { w : Iw l � p } to A. Since

lim I lw- lx l l = 0, Iw l--+oo

the basic inequality I lyo I I :::; 1 ��II�I I ' which is valid when I IY I I < 1 by virtue

of the calculations in 3.5.15, implies lim i i (w- lx) O i i = o. Thus f is Iw l--+oo

uniformly continuous on S. If n in N, the equatiori zn - 1 = 0 has as solutions the n distinct nth

. clef clef i £!!. i 27r(n - l ) roots of unIty: 1 = Wl , . . . , Wn = 1 , e n , • • . , e n For z in C and the general notation Zj �f ZWj , 1 :::; j :::; n, direct cal

culation (via induction) shows that z-nxn = (Z� lX) 0 . . . 0 (Z� lX) . n n- l The equation L Wk = 0 implies that if Rj �f - L Zj-kXk , then

k= l k= l n

Furthermore, if Z E S, then z-nxn = zj lx 0 Rj . From the identity

a o O = O o a = a


there follow

n Since L: Rj = 0,

j=l

(3.5.21 )

The uniform continuity of I in S implies that for each positive E there is a a such that a > p and I II (Pj ) - I (aj ) 1 1 < E, 1 :::; j :::; n. However, (3.5 .21 ) implies that for all n in N,

(3 .5.22)

As in earlier calculations, lim I I (a)-nxn l l = 0, whence n--+=

Because p < a, lim a-nxn = 0, whence lim (a-nxn) O = 0. Since n --+ CXJ n --+ CXJ E may be arbitrarily small (and positive) , by virtue of (3 .5 .22) ,

whence lim I Ip-nxn l i = o. n--+CXJ (3.5 .23)

1 However, since p = sr (x) , inf I I p-nxn l l -;;- = 1, and (3.5.23) yields a contranEN diction. D

[ 3.5.24 Remark. The derivation above is given in terms the properties of exp as derived in Chapter 1 . An alternative proof can be based on Liouville 's Theorem in the theory of holomorphic functions on C, cf. 5.3.29.]

3.5.25 LEMMA. IF x IS AN ELEMENT OF A COMMUTATIVE BANACH ALGEBRA A, THEN sr (x) IS sup { I pl : p E sp(x) } �f P(x) . PROOF. a) For some z in sp(x) , sr (x) :::; I z l (v. 3.5.19) , whence

sr (x) :::; P(x) .


On the other hand, if w E sp(x) and Iw l > I z l , then

Hence, if y �f w- 1x, for all large n and some t, I lyn l l � :::; t < I , and so

00

n= l

converges to yO (cf. 3.5.6c) ) . Hence w- 1x is advertible, whereas , since w E sp(x) , w- 1x, is not advertible, a contradiction:

{{z E sp(x) } 1\ {sr (x) :::; I z l }} =? { I z l = P (x)} .

If sr (x) = (1 - E )P(X) < P(x), the argument above shows

00

converges and its sum is (z- lx) O , a contradiction since z E sp(x) :

sr (x) = P(x).

[ 3.5.26 Note. Owing to 3.5.25, the notation sr (x) and the term spectral radius of x for P are justified.]

D

3.5.27 LEMMA. IF X IS A BANACH SPACE AND Y IS A CLOSED SUBSPACE, THE quotient space XjY ENDOWED WITH THE quotient norm

II I IQ : XjY :1 e r-+ inf { I lxl l : xj Y = e } IS A BANACH SPACE. PROOF. That I I I IQ i s a true norm is a consequence of the definitions and the elementary properties of inf. If {en} nEN is a Cauchy sequence in X j Y, for each n, there is an Xn such that xnjY = en and I lxn l l n I IQ + 2-n + Tm and so {xn}nEN is a Cauchy sequence. If lim Xn �f x and e �f xj Y, then lim en = e: XjY is a n4� n4� Bana�h space. D

[ 3.5.28 Note. By definition, if e = xj Y, then I le l lQ :::; I lx l l . Thus the map X :1 x r-+ xj Y �f e is norm-decreasing .]


3.5.29 DEFINITION. A left (right) ideal IN AN ALGEBRA A IS A PROPER SUBSPACE R SUCH THAT AR C R (RA C R) . A SUBSPACE THAT IS BOTH A LEFT AND RIGHT IDEAL IS AN ideal. THE IDEAL R IS regular OR modular IF THE quotient algebra AIR CONTAINS AN IDENTITY. WHEN R IS A REGULAR IDEAL AND Ul R IS AN IDENTITY IN AIR, u IS an identity modulo R. CORRESPONDING DEFINITIONS APPLY TO LEFT (RIGHT) identities modulo right (left) ideals .

3.5.30 Exercise. For a Banach algebra A: a) If u is an identity modulo the (regular) ideal R, then u is an identity modulo every ideal that contains R. b) If R is a regular left (right) ideal, then R is contained in a regular left (right) maximal ideal. c) Every maximal ideal is closed. d) An element x has a right adverse iff x is not a left identity modulo any regular right maximal ideal. e) If A ¥ Ae a proper subset S of A is a regular maximal ideal iff for some maximal ideal Me in Ae and different from A, S = A n Me.

[Hint: a) For every x, ux - x and xu - x are in R. b) Some version of Zorn's Lemma applies to the poset of ideals containing R. c) The continuity of multiplication applies . d) If x has no right adverse, then R �f { xy - y : y E A } is a right ideal, x tJ. R, and x is a left identity modulo R, and b) applies .]

3.5.31 Example. In the Banach algebra Co(ffi., q , Coo (ffi., q is a dense ideal contained in no maximal ideal: Coo (ffi., C) is not a regular ideal.

3.5.32 THEOREM. (Gelfand-Mazur) IF A IS A COMMUTATIVE BANACH ALGEBRA AND M IS A MAXIMAL REGULAR IDEAL IN A, THEN AIM IS ISOMORPHIC TO C

PROOF. With respect to the quotient norm I I I IQ , AIM is a Banach field (with identity e) and so if e i- 0, 0 tJ. sp(e) . On the other hand, sp(e) i- (/) and if z E sp( e ) , then e - ze is singular and since AIM is a field, e = ze. The correspondence e r-+ z is an isomorphism between AlM and C D

Owing to the fact that A r-+ AIM is an algebra-homomorphism, for a commutative Banach algebra A, each regular maximal ideal M may be regarded as a special element x' of the dual space A' : x' is a multiplicative linear functional. Furthermore, if x E A, then I le l l :::; I lx l l , whence I lx' l l :::; 1 .

3.5.33 DEFINITION. THE SET OF REGULAR MAXIMAL IDEALS IN A COMMUTATIVE BANACH ALGEBRA A IS Sp (A) , THE spectrum of A.

[ 3.5.34 Note. The uses of the word spectrum as in spectrum of x (when x is an element of a Banach algebra A) and spectrum of A can be misleading. However, the distinction between the two usages is clear:

• sp(x) is a set of complex numbers;


• Sp (A) is a set of regular maximal ideals .]

3.5.35 THEOREM. IF A IS A COMMUTATIVE BANACH ALGEBRA CONTAINING AN IDENTITY, Sp (A) , REGARDED AS A SUBSET OF A' , IS A WEAK'COMPACT SUBSET OF THE UNIT BALL B (0', 1 ) IN A' . PROOF. If A 3 A f-t M>. E Sp (A) is a net and M>. (J" (B', B)-converges to some x' in B (0', 1 ) , for each pair {x, y} in A,

(x, M>.) (y, M>.) = (xy, M>.) . The left resp. right member of the equation converges to (x, x' ) (y, x' ) resp. (xy, x' ) , whence x' is a continuous multiplicative linear functional: Sp (A) is weak'-closed. Owing to 3.4. 12 , B(O', 1) is weak'- compact. D

For x in A and M in Sp (A) the complex number (x, M) is denoted x(M) and the map ' : A 3 x f-t x E C(Sp (A), q is the Gelfand map. The preceding development implies that I lx(M) l loo � I lx l l . 3.5.36 Exercise. If A is a commutative Banach algebra and M E Sp (A) , then x(M) E sp(x) .

[ 3.5.37 Note. If Z E sp(x) , for some M in Sp (A), Z� lX is an identity modulo M. If y tJ. M, then fj(M) (z� lx(M) - 1) = ° and so x(M) = z. Hence I lx l loo 2: sup { Ipl : p E sp(x) } �f P(x) .

However, if M E Sp (A) and x(M) = w i- 0, then w� lx(M) = 1 and W� lX is an identity modulo M. If y �f (W� lX) O exists, then

W� lX + Y - w� lxy = 0, 1 + Y - ly = 0 ,

a contradiction: W�lX has no adverse, w E sp(x) , Iw l � P(x) :

I lxl loo = P(x) .

In C(Sp (A), q , Sp (A) is to be viewed in its weak' topology. Thus ' may be viewed as a covariant functor from the category BA., of Banach algebras containing an identity (and continuous ((>homomorphisms) to the category CF of continuous function algebras on compact Hausdorff spaces (and continuous Chomomorphisms) [Loo, Mac] .]

3.5.38 Exercise. If (J : C f-t C is a C-automorphism, then (J = id .


[Hint: If z E e, then 8(z) = 8(z · 1 ) .]

3.5.39 Exercise. If A is a commutative Banach algebra, and x E A, then

, { sp(x) x [Sp (A)] = sp(x) or sp(x) \ {O}

if A = Ae if A ¥Ae .

Since function algebras of the form C(X, q arise naturally in the study of commutative algebras, the Stone- WeierstrafJ Theorem below takes on added importance. The result is phrased in terms of the notion of a separating set of functions in a function algebra.

3.5.40 DEFINITION. A SUBALGEBRA A OF eX IS separating IFF FOR ANY TWO ELEMENTS a, b OF X, SOME f IN A IS SUCH THAT f(a) i- f(b) ; A IS strictly separating IFF FOR SOME f, f(a) = 0 = 1 - f(b) .

3.5.41 Exercise. If A is a commutative Banach algebra,

, clef A = { X : X E A }

is a strictly separating subalgebra of Co (Sp (A) , q .

3.5.42 Exercise. If A is a strictly separating sub algebra of JR.x , a, b are two elements of X, and {c, d} c JR., then, for some f in A, f (a) = c and f(b) = d.

3.5.43 Exercise. If X is a compact Hausdorff space and A is a I I 1 100-closed subalgebra of C(X, JR.) , then A is a vector lattice.

[Hint: If f E A, the Weierstrafi Approximation Theorem (3.2.24) implies that I f I is approximable by polynomial functions of f.]

3.5.44 THEOREM. (Stone-Weierstrafi) IF X IS A COMPACT HAUSDORFF SPACE, ANY CLOSED STRICTLY SEPARATING SUBALGEBRA A OF C(X, JR.) IS C(X, JR.) . (EACH STRICTLY SEPARATING SUBALGEBRA OF C(X, JR.) IS 1 1 1 i00-DENSE IN C(X, JR.) . )

PROOF. If f E C(X, JR.) and a , b are two points in X , A contains an fa,b such that fa,b (a) = f(a) , fa,b (b) = f(b) . If E > 0, then

Uab �f { x : fab (X) < f(x) + E } and Vab �f { x : fab (X) > f(x) - E }

are open. If b is fixed, {UabLEx is an open cover of X. Hence there is a finite subcover {Ua1 b , . . . , Uapb} and, owing to 3.5.43, fb �f inf fa b E A. l�p�P p If x E X, for some p, x E Uapb and fb (X) < f(x) + E. On the other hand, if


p x E Vb �f n Vapb , then Ib (X) > I(x) - f . The open cover {Vb hEX admits

p=l

a finite subcover {Vb" . . . , VbQ } and 3.5.43 implies ¢ �f sup Ibq E A. If l�q�Q

x E X, for some q, x E Vbq and I(x) - f < ¢(x) < I(x) + Eo D [ 3.5.45 Note. a) If A is merely separating, A can fail to be C(X, JR.) , e.g. , if X = [0, 1] ' the set A of all polynomial functions that vanish at zero is separating. However ,

A = C(X, JR.) n { I : 1(0) = O } .

If JR. is replaced by C in the discussion above, the corresponding conclusions are false, v. Chapter 5. What is true and what follows directly from 3.5.44 is that if A is a closed strictly separating subalgebra of C(X, JR.) and 1 E C(X, q , then 1 = u + iv , {u, v } C C(X, JR.) and by abuse of notation, C(X. q = A + iA. There is a corresponding statement if A is merely separating.]

3.5.46 Exercise. If X is a compact Hausdorff space and a subset A of C(X, JR.) is both A-closed and v-closed, the II l loo-closure of A contains each continuous function approximable on every pair of points by a function in A.

3.5.47 Exercise. a) The set

is strictly separating. b) The smallest algebra A over 1I' and containing <t> is the JR.-span of <t>. c) The algebra A + iA is I l l i oo-dense in C(1I', q. d)

2� ,

The map I : C(1I', JR.) 3 1 r-+ 1 1 (eix) dx is a DLS functional (with an associated measure T) . e) For the maps

W : [0 , 27r] 3 x r-+ eix E 1I', W* : C1I" 3 1 r-+ l o W E C[O,2�l , W* [L2 (1I', T)] = L2( [0, 27r] , A) and 1 1 1 1 1 2 = I IW* (f) 1 1 2 ' f) The set W* (<t» is a CON in L2 ( [0 , 27r] , A ) . 3.5.48 Exercise. For a commutative Banach algebra A such that A = A e , i f x E A: a) { x(M) : M E Sp (A) } is a compact subset of C; b) for the closure A(x) of the set of all polynomials

n aoe + L akxk , ak E C, n E N,

k=l


the map h : Sp [A(x)] 3 M r-+ x(M) E C is a homeomorphism.

3.6. Hilbert Space

The groundwork for studying Hilbert space S) was laid in the material starting with 3.2 .9 and ending with 3.2.18. The function ( , ) is a positive definite and conjugate bilinear form. The current Section is devoted to deriving some fundamental results about the character of operators , i.e. , for various Banach spaces X, elements of [S) , X]c .

3.6.1 THEOREM. (F. Riesz) IF x' E S)' , FOR SOME x IN S),

(y, x) == (x, x') . [ 3.6.2 Remark. In the preceding sentence there is a possibility for notational confusiOn: the symbol (y, x) in the left member of the identity denotes the value of the inner product in S); (x, x') in the right member denotes the value of the functional x' acting on the vector x. The content of the THEOREM and the nature of the PROOF should clarify any distinction.]

PROOF. The kernel M �f { z : (z , x' ) = O } is a closed subspace of S); Zorn's Lemma implies that M contains a maximal orthonormal subset M �f {m),} ), EA " If M = S), then 0 serves for x. If M ¥S) and v E (S) \ M) ,

then (v, x' ) -j. o. Moreover, U �f -( v

) E S) \ M and (u, x') = 1 . For any v, x' w, m �f w - (w , x' ) u E M. Thus, if z �f u - L (y, a), ) a)" then z -j. 0,

), EA

Z E M�, and if a �f 1 1: 1 1 ' then { { a}l:.!M} is a CON set in S). Hence, if

x �f (a, x')a, for any y, y = (y, a)a + L (y, m),) m), and ),E A

(y, x' ) = (y, a) (a, x') = (y, (a, x' )a) �f (y , x) . D

3.6.3 Exercise. The correspondence S)' 3 x' r-+ x E S) is a conjugatelinear bijection. If {a, b} C S), then (a, b) = (a' , b' ) .

In view of the conjugate character of the bijection, S) and S)' are essentially the same. 3.6.4 Exercise. If {U, V} C [S)] c and {x, y} C S), then

{ (U(x) , x) == [x, V(x)] } {} { (U(x) , y) == [x, V(y)] } .

Section 3.6. Hilbert Space

[Hint: (U(x ± iy) , x ± iy ) = [x ± iy , V(x ± iy ) ] . The replacement technique x r-+ x ± iy is known as polarization .]

129

3.6.5 DEFINITION. WHEN S C SJ, THEN S� �f { x : (S, x) = {O} } . 3.6.6 Exercise. a ) For any S, S� i s a closed subspace. b) If M is a closed subspace and x E SJ there is in ]\,{ x M � a unique pair {a, b} such that x = a + b: SJ = M ffi M�. c) the projection PM : SJ 3 x r-+ a E M is in [SJk d) pk = PM and I IPM II = l .

[Hint: A maximal ON subset M �f {ill>. }),EA of ]\,{ is contained in a CO� set H for SJ and a = L (x, ill), ) ill), .] ),EA

3.6.7 DEFINITION. FOR T IN [SJ]e , THE kernel OF T IS

AND THE image OF T IS im (T) (�f T(SJ) .

f) id + TIT; IS INVERTIBLE. PROOF. The results a)-d) follow from direct calculation.

e) On the one hand,

I ITI T{ I I = sup I ITIT{ (x) 1 1 � I ITI I I sup I IT{ (x) 1 1 1 � I ITI I I · I IT{ I I = I ITI I 1 2 , I lx l /= 1 I /xl /= I and on the other hand, if I l xl l = 1 , then

I ITIT; (x) 1 1 ;? I (TIT{ (x) , x) 1 = I IT{ (x) 1 1 2 . Thus I ITIT{ I I ;? I IT{ 1 1 2 = I ITl I 1 2 .

f) Since ( [id + TIT{] (x) , x) = I I xl 1 2 + I IT{ (x) 1 1 2 ;? I lx 1 1 2 , it follows that ker (id + TIT; ) = {O}. Thus im (id + TIT; ) is dense in SJ. If x E SJ, there is a sequence {Yn}nEN such that lid + TIT;] (Yn ) converges to x. Because id + TIT; is norm-increasing, {Yn}" EN is a convergent sequence, and if


lim Yn �f y, then lid + TIT{ ] (y) = x: id + TIT{ is bijective and thus is n---+= invertible. D

[ 3.6.9 Note. When dim (SJ) < 00, every injective element of [SJ] is invertible. By contrast, 5 in 3.5.8 is injective and is not invertible. An invertible operator T is not only injective but also bijective.]

3.6.10 Exercise. If T E [SJ]c , then ker (T) = lim (T,)] -L (cf. 3.4. 14) . [Hint: { (T(x) , y) == O} {} { (x, T' (y) ) == O} .]

3.6 .11 DEFINITION . A T IN [SJ]c IS self-adjoint IFF T' = T; T IS normal IFF TT' = T'T. 3.6.12 Exercise. If T E [SJ]c , then T + T' and i (T - T') are self-adjoint; if T is normal, T + T' and T - T' commute.

3.6.13 THEOREM. (The Spectral Theorem) IF T IS NORMAL , THE (COMMUTATIVE) BANACH ALGEBRA AT GENERATED IN [SJ]c BY {id , T, T'} IS , VIA THE MAP ' isometrically isomorphic TO C (Sp (AT) , q . PROOF. The map � is linear (v. 3.5.28) . Owing to 3.5.37, for any . I I � I I . 5 + 5' . 5 - 5' def . . 5 m AT, sr (5) = 5 = . Smce 5 = -2- + l -z;;- = U + lV, 5 IS a linear combination of the self-adjoi�t operators U and V. Furthermore, 5' = U' - iV' and §t = u' - {0 = 5 (cf. 3.4.16) . Hence

55' = UU' - VV' + i (VU' + U'V) , 1 1 55' 1 1 = 1 1 5 1 1 2 = 1 1 52 1 1 (cf. 3.6.9) ,

1 1 5 1 1 = 1 1 52 1 1 � = . . . = 1 1 52n I ITn -t sr (5) = 1 15 1 1= , i .e. , � is an isometry and hence (v. 3.4.2 ) AT is a closed subspace of C (Sp (AT) ' q .

If M E Sp (AT) , 5 is a self-adjoint element in AT, S(M) �f a + ib, a E JR., and b -j. 0, for L �f 5 + iO'id ,

a2 + b2 + 2bO' + 0'2 ::::: I lt l !: ::::: I I LI 12 = I I LL' I I = 1 1 52 + 0'2 1 1 ::::: 1 1 5 1 1 2 + 0'2 .

Thus a2 + b2 + 2bO' ::::: 1 1 5 1 1 2 , which is false if O'b is sufficiently large. Hence b = O: S(M) E R

By definition, if MI and M2 are two elements in Sp (AT) ' for some self-adjoint 5 in AT, S (Ml ) -j. S (M2 ) and a direct calculation shows that


fj �f { S : S a self-adjoint element of AT } is strictly separating. The Stone-Weierstrafi Theorem implies fj is dense in C (Sp (AT) , JR.) . Hence fj + ifj (= AT) is a dense (v. 3.5.45) and, as shown above, a closed subspace of C (Sp (AT) , q . D

[ 3.6. 14 Remark. The isometric isomorphism established yields an interpretation to be given to f (T) when f E C(Sp (A) , q . This observation is the basis for a functional calculus for normal operators. Furthermore, T : Sp (A) 3 M r-+ T(M) E ((: is a continuous injection that permits the identification of Sp (A) with a compact subset sp(T) , the spectrum of T, in C. ]

3.6. 15 DEFINITION. A LINEAR MAP T : S h r-+ Sh (T E [S h , Sh ] ) BETWEEN HILBERT SPACES Sh resp. Sh ENDOWED WITH INNER PRODUCTS ( , ) resp. [ I ] IS : a) unitary IFF T (SJ t ) = SJ2 AND (x, y) == [Tx ITy] ; b) AN isometry IFF II xii == I ITx l l · 3.6. 16 Exercise. a) A unitary operator is an isometry. b) An isometry need not be unitary. c) If T E [SJ] and T is unitary, then T is normal and bijective. Furthermore, T' = T� l . d) If SJ �f ((:2 endowed with the inner product 2 ( , ) : (((:2 ) 3 { (Xl , X2 ) , (YI , Y2 ) } r-+ XIYI + X2Y2 and T in [SJ] is such that

then T is unitary (hence normal) but T is not self-adjoint . [Hint: For b) , 3.5.8 applies .]

3. 7. Miscellaneous Exercises

3.7. 1 Exercise. In the context of Young's inequality (3.2 .1 ) , if c 2: 0, then

lc l<1>(C) o ¢(s) ds + 0 '1jJ(S) dS = A ( [O, C] x [O, ¢(c)] ) = c¢(c) .

(The equation above implies Young's inequality without recourse to arguments based on geometry. ) �.7.2 Exercise. If (X, S, f.-l) is a measure space and f E 5, then

(3.7.3)


iff for some a in JR., sgn (I) . f � eia . f, i.e. , I f I = eia f.

[Hint: If (3.7.3) holds, A �f Ix f dp, -j. 0, and sgn (A) = eia , for

eia f �f g �f h + ik, it follows that

Hence

Ix g dp, = Ix h dp, + i Ix k dp, = Ix eia f dp,

= Ilx f dP, 1 = Ix l f l dp,.

Ix g dp, = Ix h dp, � Ix I h l dp,

� lx (h2 + k2) � dp, = Ix g dp"

k � 0, Ix g dp, = Ix I g l dp" and a.e. ,

° � g = eia f = l eia f l = If I = sgn (I)f a.e.]

3.7.4 Exercise. If BI and B2 are Banach spaces and T E [BI ' B2] , then T is continuous iff for some S in [B; , B� ] , each Xl in BI , and each y; in B; :

in which case, S = T', i.e. , T is continuous iff T' exists. [Hint: The Closed Graph Theorem (3.3.18) applies .]

3.7.5 Exercise. a) If 1 � p < 00, the set S of simple functions is dense in LP(X, p,) metrized by II l ip · b) For some (X, 5, p,) , S is not dense in LOO(X, p,) . c) If 1 � p < 00 and LP is derived from a DLS functional I defined on a function lattice L, then L n LP is a dense subset of LP . d) For some function lattice L and some DT Q functional I defined on L, L n L 00 is not dense in L 00 . 3.7.6 Exercise. If X is a compact Hausdorff space and A is a II 1 1 00-closed subalgebra of C (X, JR.) , then A is closed with respect to the lattice operations 1\ and V.

[Hint: The relation between the lattice operations and I I and the Weierstrafi Approximation Theorem (3.2.24) apply.] For f in L2 ( [0 , 1] , A) the sequence of its Fourier coefficients is

{cn �f _1_ t f(t)e-2nrrit dt} . v'2ii Jo nEZ

Section 3.7. Misoellaneous Exercises

The following relevant functions figure in the Exercises below.

N S (t) �f "" _1_ 2nrrit N � Cn � e ,

n=-N V 271"

clef 1 sin (2N + 1) 7I"t DN(t) = - t 2 (Dirichlet's kernel) , 271" • 71" Sln -2

F (t) �f 1 (FeJ' er 's kernel) . [sin ( (N � 1 )7I"t ) ] 2

N - 271"(N + 1) sin (�t )

133

1 N 3.7.7 Exercise. a) FN = -- "" Dn. b) DN and FN are periodic and N + 1 �

n=O each is of period one. c)

11 Dn dt = 1

1 FN dt = 1 ,

SN(t) = 11 DN(t - s)f(s) ds �f DN * f(t ) ,

N 1 1 "" r clef N + 1 � SN(t) = in Fn(t - S)f(8) ds = FN * f(t ) . n=O 0

(3.7.8)

1 I U d) If 0 < E < 2 ' then FN [0, 1]\ [< , 1 - <] ---+ 0 as N ---+ 00 . e) If f E C( [O, 1] , q and f(O) = f(l ) , then lim I I FN * f - fl loo = 0 (Fejer 's Theorem) . N---+oo

[Hint: For e) , if E E (0, � ) , since FN 2': 0, IFN * f(t) - f(t) 1 � 1

< + 1

1 - < + 1� < FN(t - s) l f(s) - f(t) 1 ds

�f I + I! + I! I.

Because f is bounded, if N is large, then I + II is small since FN � 0 on [0, E] U [1 - E, 1] . If E is near � , since f is continuous, 2 (3.7.8) and the nonnegativity of FN imply II! is small.]

3.7.9 Exercise. Fejer's Theorem (3.7.7e)) implies the Weierstrafi Approximation Theorem (3.2.24) .


[Hint: If I E C( [O, 1] , ffi.) , in C( [O , 27r] , ffi.) there is an 1 such that 11 [0, 1 ]= I while !(27r) = 1(0) . A change of scale applies .]

3.7. 10 Exercise. a) If 1 :s; p < 00 I E LP ( [O, 1] , ), ) , then

lim I IFN * I - II I = 0. N---+= P [Hint: The results 3.7.7e) and 3.7.9 apply.] For a measure space (X, 5, p,), a sequence {In} nEN of measurable func

tions converges in measure to I (In m-=ts f) iff for each positive E ,

lim p, ( { x : l in ( x) - I ( x ) I > E }) = 0. n---+=

3.7. 11 Exercise. If (X, 5, p,) is a measure space, each In increases monotonely, In m-=ts I, and I is continuous at x (x E Cont (I) ) , then

lim In(x) = I(x) . n---+=

[Hint: If h is small, for some J(h) , I(x) = I(x + h) + J(h) while I J (h) 1 is small. If E > 0, h > 0, and for infinitely many n,

In (x) - I(x) 2': E,

then for infinitely many n,

In(x + h) - I(x + h) 2': In (x) - I(x) + J(h) ? E + J(h) .

If h i s small, E + J (h) 2': � . Hence, for infinitely many n ,

{ y : In (Y) - I(Y) }

contains a fixed interval of posi ti ve measure. If I ( x) - In (x) 2': E for infinitely many n, the argument applies for I h l small and h negative.]

3.7.12 Exercise. If (X, 5, p,) is a totally finite measure space, I E S, I is strictly positive a.e. (p,), and for a sequence {En} nEN of measurable sets, lim r I dp, = 0, then lim p, (En ) = 0. n--+CXJ } En n--+CXJ

[Hint: p, ( { x : I(x) > � } ) t p,(X). ]


3.7. 13 Exercise. (Schwarz) If 15 is a finite set and {aa}aE8 and {ba}aE8 are two sets of complex numbers, then

Equality obtains in the preceding inequality iff for some A and B, not both zero, Aaa = Bba , Cl' E 15 .

[Hint: 3.2.10 applies.]

3.7. 14 Exercise. (Fischer-Riesz) If {xa}aEA is a CON set in S) and X E S), then L l (x, xa ) 12 < 00. Conversely, if L laa l 2 < 00, for the poset aEA aEA b.(A) of all finite subsets of A, the net n : b.(A) 3 15 r-+ L aaxa converges

to some x in S) and for each Cl' , (x, xa ) = aa . [Hin" If ' E �(A) , then I I� aaxa l l

' � � laa l ' ·]

3.7. 15 Exercise. If {xa}aE8 is a finite orthonormal set and X E S), then

[Hint:

3.7.16 Exercise. (Holder's inequality extended) If

I 1 1 < Pi , 1 :::; i :::; I, L - = 1 , i=l Pi

I I and Ii E LPi (X, p,) , then I �f II Ii E Ll (X, p,) and 1 1 1 1 1 1 :::; II I l li l lpi ·

i=l i=l

[Hint: If PI �f p, then p' = (t �) - 1 and induction applies .]

i=2 Pt


3.7. 17 Exercise. If (X, S, I-l) is a measure space, f E S, and

A �f { p : 0 < p < 00, I l f l lp < oo } :

a) A is convex; b) A 3 P r-+ II f l ip is continuous; c)

{r < p < s} =? { l lf l lp � max { l lf l l r , II f i l s} }

(regardless of whether any of r, p, s is in A) ; d)

{A ¥= 0} =? { lim II f l ip = I l f l loo } ; p---+oo

e) when X E S and I-l(X) < 00, {p < q} =? { l lf l lp � I l f l l q} . 3.7.18 Exercise. For each x in [0, 1] , the value Bn(f) (x) of the nth Bernstein polynomial is a convex linear combination of the values of f at the points '5.- , 0 � k � n. n 3.7.19 Exercise. a) If X is a Hausdorff space B is the set of bounded continuous JR.-valued functions on X, with respect to the norm II 1 1 00 , B is a Banach space. b) For x in X, the evaluation map �x : B 3 f r-+ f (x) is an element of B' and I I �x l l � 1 . c) The map I : X 3 x r-+ �x E B' is a continuous injection. d) The weak' closure ,6(X) of I (X) is weak'-compact. (The space ,6(X) is the Stone-tech compactijication of X.) e) If F E C(X, JR.) , for some F in C(,6(X) , JR.) , F lx= F: each continuous function on X has a continuous extension to ,6(X) , cf. 1 .7.28.

A subset S of a topological vector space V is bounded iff for every neighborhood N of 0 and some real A, S C AN.

3.7.20 Exercise. (Kolmogorov) A Hausdorff topological vector space (V, T) is normable, i .e. , T is norm-induced by some norm I I I I , iff there is in V a bounded convex neighborhood of O.

[Hint: If V i s normable, then N �f { x : I l xl l � I } is a bounded convex neighborhood of O.

If N i s a bounded convex neighborhood of 0, then N contains a circled neighborhood U for which Conv (U) is bounded. The Minkowski functional pu is a norm that induces T.]

4 More Measure Theory

4.1. Complex Measures

The Riesz Representation Theorem 2.3.2 yields a measure space (X, 5, p,) and an integral representation of a nonnegative element in [Co (X, ffi.)] ' . The more general problem of finding an integral representation of arbitrary elements in [Co (X, ffi.] ' resp. [Co (X, C] ' deserves a solution. It is given in Section 4.3.

The measure in 2.3.2 takes values in [0, 00] ; for

[Co (X, ffi.)]' resp. [Co (X, q]"

one might expect measures taking values in ffi. resp. C. 4.1.1 Example. For the measure space (X, 5 , p,), when f E Ll (X, p,) , the map � : 5 3 E r-+ Ie f dp, is a C-valued set function. If {En} nEN is a se-

quence of pairwise disjoint sets in 5 and E �f U En , the results in nEN Chapter 2 imply:

� is a countably additive C-valued set function such that � (0) = o. The situation just described motivates the following development.

4.1.2 DEFINITION. FOR A SET X AND A a-RING 5 CONTAINED IN !fj(X) A SET FUNCTION � : 5 3 E r-+ � (E ) E C IS A complex measure IFF: a) � (0) = 0; b) FOR A SEQUENCE {En}nEN OF PAIRWISE DISJOINT SUBSETS IN 5 ,

00 � (U nENEn) = L � (En ) (� IS COUNTABLY ADDITIVE) . IN THE CIR-

n=l CUMSTANCES DESCRIBED, (X, 5 , 0 IS A complex measure space .

137

138 Chapter 4. More Measure Theory

[ 4.1 .3 Remark. Henceforth, according as 1-l(5) C [0, 00] ' resp. 1-l(5) C [-00, (0) or 1-l(5) C (-00, 00]' resp. 1-l(5) C C, (X, 5, I-l) is a measure space, resp. a signed measure space, resp. a complex measure space . For a signed measure space (X, 5 , I-l) , the possibility 1-l(5) = [-00, 00] is excluded. Otherwise, there arises the awkward question of assigning a value to 00 - 00 or -00 + 00.]

4.1.4 Exercise. For a set X and a a-ring 5 contained in !fj(X) , a set function � is a complex measure iff: a) for every E in 5 and every countable 00 measurable partition {En}nE]\j of E, � (U nE]\jEn) = L � (En) and b) for

n=l 00 every countable measurable partition {En}nE]\j, L I� (En) 1 < 00.

n=l 4.1.5 Exercise. If (X, 5, 0 is a complex measure space and for E in 5 , O'(E) �f �[�(E)] , (3(E) �f <;S[�(E)] , then a and (3 are complex measures and 0'(5) U (3(5) C R 4.1.6 Exercise. For (X, 5, I-l) , if 1-l(5) C [0, (0) , then

1] �f sup { 1-l(E) : E E 5 } < 00.

[Hint: For some sequence {En}nE]\j, En C En+l and

4.1.7 Exercise. For (X, 5, I-li ) , if 1-l; (5) C [0, (0) , 1 � i � 4, then

is a complex measure.

4.1.8 DEFINITION. FOR A SIGNED MEASURE SPACE (X, 5 , I-l) , A SET A IS A positive set (negative set ) IFF FOR EACH E IN 5 , (E n A) E 5 AND I-l (E n A) � 0 (I-l (E n A) � 0 ) .

For � in 4 .1 .1 , any set where ! is nonnegative is a positive set. If A C B and ! is positive on B (hence also on A) I-l [E> (I, 0) ] > 0, then Ed!, O) is a negative set and E> (1, 0) is a positive set. If Ix ! dl-l < 0, then 4.1 .9 Exercise. a) Every measurable subset of a positive set is a positive set. (Hence, if El and E2 are positive measurable sets, then El \ E2 and

Section 4.1 . Complex Measures 139

EI n E2 are positive measurable sets. ) b) If {En} nEN is a sequence of pairwise disjoint positive sets, U En is a positive set. c) If E is a nEN positive set and A E 5, then E n A is a positive set. d) Similar assertions obtain for negative sets.

4.1 .10 THEOREM. (Hahn) IF (X, 5, p,) IS A SIGNED MEASURE SPACE AND

p,(5) C [-00, (0) or p,(5) C (-00, 00] ,

THEN FOR SOME P IN 5 AND Q �f X \ P AND EACH E IN 5 , P n E, Q n E E 5 , p,(P n E) 2': 0, p,(Q n E) :::; 0 ,

p,(E) = p,(P n E) + p,(Q n E) .

PROOF. The argument below is given when p,(5) C [-00, (0) . A similar argument is valid when p,(5) C (-00, 00] .

If 1] �f sup { p,(E) : E E 5 , and E is a positive set } , then 1] < 00 (v. 4.1 .6) . If 1] = 0, then (/) serves for P. If 1] > 0 there is a sequence {En}nEN of measurable positive sets such that p, (En) t 1]. If P �f U En , then

P = EILJ U �=2 (En \ En- I ) E 5 and for each M in N,

00 1] 2': p,(P) = p, (EM) + L P, (En \ En-d 2': P, (EM) '

n=M+I whence p,(P) = 1]. If E E 5, then

00 p,(P n E) = P, (EI n E) + L p, [(En \ En-d n E] 2': 0 :

n=2 P is a positive set.

If Q �f X \ P and Q is not a negative set a contradiction is derived by the following argument .

For some measurable Eo contained in Q, 00 > P, (Eo ) > o. But Eo is not a positive set, since otherwise, PLJEo is a positive set and

p, (PLJEo) = p,(P) + p, (Eo ) > 1], a contradiction. Hence, for some measurable subset E of Eo , p,(E) < o. In

1 1 1 the sequence -1 , - - , - - , . . . there is a first, say - - , such that for some 2 3 ml 1 measurable subset EI of Eo, p, (Ed < - - . Then ml

p, (Eo \ Ed = p, (Eo ) - p, (Ed > p, (Eo ) > o.


The argument applied to Eo now applies to Eo \ EI : for some least positive integer m2 and some measurable subset E2 of Eo \ EI , P, (E2 ) < - � and m2 by induction there is a sequence {En} nEN of pairwise disjoint sets and least positive integers {mn } nEN such that

1 p, (E2 ) < -- , m2 p, [Eo \ (EI U E2)] = p, (Eo ) - p, (EI ) - p, (E2 ) > 0,

ex:> 1 Hence L - < 00 and mk -+ 00 as k -+ 00 . The earlier argument n=l mn

implies that A �f Eo \ U En is not a positive set. Hence, for some nEN 1 measurable subset F of A and some mK , mK > 2 and p,(F) < - -- . But mK

then F U EK c Eo \ (u ::11 Ek) and

imply a contradiction of the minimality of mK : Q is a negative set and the pair {P, Q} performs as asserted. 0

[ 4.1 .11 Remark. The pair (P, Q) is a Hahn decomposition of X. Although as constructed, P E.S , i f X tJ- 5, then Q tJ- 5.]

4.1 .12 Exercise. If (Pi , Qi ) , i = 1 , 2, are Hahn decompositions of X for the signed measure space (X, 5, p,) , then for any measurable set A,

p, (A n PI ) = p, (A n P2)

(and hence p, (A n QI ) = p, (A n Q2) ) ' [ 4.1 .13 Note. If p,(N) = 0 and N \ P -j. 0, then a second Hahn decomposition is (P U N, Q \ N) . Examples of such N abound. For example, if f �f X [ l and p, in (JR., 5)" p,) is defined by 0, 1

p,(E) �f Ie f dx,

Section 4.1 . Complex Measures

then P �f [0, 1] is a positive set and Q �f ffi. \ { [O, I] } is a negative set and (P, Q) is a Hahn decomposition. If N �f Q, then IAN ) = 0, N \ P -j. 0, and (P U Q, Q \ Q) is a second Hahn decomposition. Thus Hahn decompositions are not necessarily unique but they all produce the same effects.]

141

4.1 .14 DEFINITION. WHEN (X, 5, f.-l) IS A SIGNED MEASURE SPACE AND (P, Q) IS A HAHN DECOMPOSITION FOR f.-l, THEN FOR E IN 5 ,

( clef ( ( clef ( f.-l+ E) = f.-l E n P), f.-l- E) = -f.-l E n Q) .

[ 4.1 .15 Remark. In light of 4 .1 .12 , the set functions f.-l± are independent of the choice of (P, Q ) .]

4. 1 . 16 Exercise. The set functions f.-l± are (nonnegative) measures.

4. 1 .17 THEOREM. IF (X, 5, �) IS A COMPLEX MEASURE SPACE, THE SET FUNCTION

I � I : 5 3 E r-+

sup { � I� (En ) 1 : {En}nEN a measurable partition of E } IS A MEASURE AND 1 � 1 (5) C [0, (0) .

PROOF. Since the only measurable partition of 0 is itself, 1 � 1 (0) = 0. If {En} nEN is a measurable partition of E, then for each n in N and

each positive E, En admits a measurable partition {Enk hEN such that 00 '" I� (Enk ) 1 � I � I (En ) - 2: ' Since E = U Enk , it follows that � {k ,n}CN k=1

00

{k ,n}CN 00

whence IWE) � L I � I (En) : I� I is superadditive. n=1 .

On the other hand, the partition {En} nEN may be chosen so that

00 00

n=1 n=1


whence I � I is countably additive: I � I i s a measure. The set functions a �f � [�] , (3 �f �[�l are signed measures and

0'(5) U (3(5 ) c ffi..

Corresponding to the four measures o'± and (3± there is the Jordan decomposition � = 0'+ - 0'_ + i ((3+ - (3- ) of �. Since the ranges of the measures o'± , (3± are contained in [0, (0) , it follows that for E in 5 and in the context above,

00 00

n=1 n=1 The definitions of o'± , (3± imply that

M �f sup { O'+ (D) + O'_ (D ) + (3+ (D) + (3- (D) D E 5 } < 00,

whence IWE) :::; M. D 4.1.18 Exercise. If (X, 5, f.-l) is a signed measure space, then

f.-l = f.-l+ - f.-l- and I f.-l l = f.-l+ + f.-l- .

4.1.19 DEFINITION. IF (X, 5 , �) IS A COMPLEX MEASURE SPACE,

Ll ( ) clef 1 ) 1 ) X, � = L (X, o'± n L (X, (3± ,

4.2. Comparison of Measures

When (X, 5, f.-l) and (X, 5 , �) are measure spaces or complex measure spaces the relation between f.-l(E) and �(E) as E varies in 5 deserves analysis. Examples of such a relation are: a) 0 :::; f.-l(E) + �(E) < 00; b) IWE) :::; f.-l(E) ; c) (more generally) Ll (X, O C L1 (X, f.-l) . Although each of the preceding is of some interest, the relations that have emerged as of fundamental importance are those given next .

Section 4.2. Comparison of Measures 143

4.2 .1 DEFINITION. FOR MEASURE SPACES (X, 5, I-l) AND (X, 5, � ) , I-l IS absolutely continuous WITH RESPECT TO � (I-l « 0 IFF

{�(E) = O} =? {1-l(E) = O} ; WHEN (X, 5, I-l) AND (X, 5 , 0 ARE COMPLEX OR SIGNED MEASURE SPACES, I-l � � IFF { IWE) = O} =? {1-l(E) = O} . WHEN

AIL E 5 AND I-l(E) == I-l (E n AIL) ,

I-l lives ON AIL' WHEN I-l LIVES ON AIL ' � LIVES ON A� , AND AIL n A� = 0, I-l AND � ARE mutually singular (I-l -1 � ) . 4.2.2 Example. If (X, 5, I-l ) is a measure space, I is nonnegative and 5-measurable, and, for each E in 5, �(E) �f Ie I dl-l, then � « I-l. 4.2.3 Example. For the map I-la : 5), 3 E r-+ A (E n Ca) , ( [0, 1] , 5)" I-la) is a measure space and I-la lives on Ca. If a = 0, then A lives on [0, 1] \ Ca and I-lo -1 A.

Although the circumstances just described exemplify the relations

� « I-l and � -1 I-l, the following discussion provides a more refined sorting out of the possibilities for those relations.

4.2.4 THEOREM. IF (X, 5, I-l) AND (X, 5, �) ARE TOTALLY FINITE MEASURE SPACES , THEN FOR SOME MEASURES I-la AND I-ls: a) I-l = I-la + I-lsi b) I-la « I-l; c) I-ls -1 I-l; d) I-la -1 I-ls ; e) FOR SOME NONNEGATIVE INTEGRABLE h, AND EACH E IN 5, l-la (E) = Ie h d� . PROOF. (von Neumann) The following argument derives a)-e) more or less simul taneously.

If p �f I-l + � and I E L2 (X, p) , then I E L2 (X, I-l) n L2 (X, � ) . Owing to Schwarz's inequality (3 .2 . 1 1 ) ,

1

Ii I dl-l l � i III dl-l � i III dp � (i 1 1 1 2 dP) "2 • [p(X)] � < 00.

Hence T : L2 (X, p) 3 I r-+ T(f) �f i I dl-l E C is in [L2 (X, p)] ' . Riesz 's result (3.6. 1) implies that L2 (X, p) contains a g such that for

every E in 5 and every I in L2 (X, p) , Ie dl-l = i XE dl-l = L XEg dp = Ie g dp, (4.2.5)

L(l - g)l dl-l = i lg dP - i lg dl-l = i lg d�. (4.2.6)


If p(E) > 0, (4 .2 .5) and the inequality 0 :::; f.-l(E) :::; p(E) imply

0 :::; p(�) · Ie g dp :::; 1 .

Thus g ? 0 a.e. and p [E> (g , 1 ) ] = 0: 0 :::; g (x) :::; 1 a.e. Modulo a null set (p) , X = { x : 0 :::; g (x) < l } l:J { x : g (x) = 1 } �f Al:JS . Thus, for E in 5 ,

f.-l(E n A) + f.-l(E n S) �f f.-la (E) + f.-ls (E), f.-la « f.-l, f.-ls -1 f.-l.

For n in N, E in 5, and f �f (� gk) . XE, (4.2.6) reads

(4.2.7)

As n -+ 00, if E C A, the left member of (4 .2 .7) converges (by virtue of Lebesgue's Monotone Convergence Theorem) to f.-la(E) and the integrand in the right member of (4.2.7) converges monotonely to some h: for any E in 5 , f.-la (E) == Ie h d�. In particular, 0 :::; h E Ll (X, � ) . D

[ 4.2.8 Remark. The equation f.-l = f.-la + f.-ls represents Lebesgue 's decomposition of f.-l: f.-la is the absolutely continuous component and f.-ls is the singular component of f.-l. The equation

f.-la (E) == Ie h d�

is the expression of the Radon-Nikodym Theorem. The complex of results and assertions is sometimes referred to as the LebesgueRadon-Nikodym Theorem or LRN. The function h is the Radon-Nikodym derivative of f.-l with respect to � : h = df.-l .] d�

4.2.9 Exercise. The Radon-Nikodym derivative of f.-l is unique modulo a null function (f.-l) . 4.2 . 10 Exercise. The validity of LRN persists if f.-l is a complex measure and if X is totally a-finite (with respect to If.-l l ) . 4.2 .11 Exercise. If (X, 5 , f.-l) and (X, 5 , �) are measure spaces such that X E 5 and f.-l(X) + �(X) < 00 ( (X, 5 , f.-l) and (X, 5, O are totally finite) ,

Section 4.2. Comparison of Measures 145

� « p" and � is not identically zero, then for some positive E and some E in 5, E is a positive set for � - Ep,.

[Hint: For each n in N, if (Pn, Qn) is a Hahn decomposition for � - �p" then � ( n Qn) = ° < � (U Pn) . For some no ,

nEN nEN 1 p, (Pno ) > 0, E = Pno , and E = -.] no

4.2.12 Exercise. If X in (X, 5, p,) is totally finite and � « p" then: a)

A �f { f : O :::; f E Ll (X, p,) , M(E) �f sup r f dP, :::; � (E) } -I 0; EES JE

b) for some nonnegative h in S and each E in 5, l h dv = M(E) ; c)

�(E) == l h dp, (hence h = ��) . (The preceding conclusions yield a second proof of LRN.)

[Hint: For c) , 4.2 . 11 applies.]

4.2.13 Exercise. If (X, 5, p,) is totally finite, � « p" d� = h, and g is in dp, LOO (X, � ) , then 1 9 d� = 1 9 h dP,.

4.2.14 THEOREM. IF (X, 5 , p,) IS A MEASURE SPACE AND (X, 5 , O IS A COMPLEX MEASURE SPACE, THEN � « p, IFF FOR SOME MAP

J : (0, (0) 3 E r-+ J(E) E (0, (0) ,

THE IMPLICATION {p,(E) < J(E)} ::::} { I�(E) I < E} OBTAINS.

PROOF. If � « p, and no J as described exists, then for some positive E, each n in N, and some En in 5, p, (En) < Tn while I� (En) 1 2': E . Then p, ( lim En) = ° while I� I ( lim En) 2': E, a contradiction. n4� n4�

If J as described exists, p,(E) = 0, and E > 0, then 1p,(E)1 < J(E ) , and thus I�(E) I = 0. D

4.2 .15 Exercise. If (X, 5, p,) is a measure space, h E L1 (X, p, ) , and for E in 5 , �(E) �f l h dp" then �� = h and IWE) = l l h l dp,.

[ 4.2.16 Remark. In the preceding discussion there are references to various special measure spaces, e.g. , a (X, 5, p,) that is totally finite.


For a given measure space (X, 5 , p, ) , the following classification of possibilities is useful. A measure space in one class belongs to all the succeeding classes. a) (X, 5, p,) is totally finite, i.e. , X E 5 and p,(X) < 00;

b) (X, 5 , p,) is finite , i.e. , for each E in 5, p,(E ) < 00;

c) (X, 5 , p,) is totally a-finite, i.e. , X is the union of count ably many measurable sets, each of finite measure;

d) (X, 5, p,) is a-finite, i .e. , each E in 5 is the union of countably many measurable sets, each of finite measure;

e) (X, 5 , p,) is decomposable, i.e. , 5 contains a set F of pairwise disjoint elements F of such that: e1) X = U F; e2) for each

FEF F in F, p,(F) is finite; e3) if p,(E ) is finite,

p,(E ) = L p,(E n F), ( 4.2. 17) FEF

(whence there are at most count ably many nonzero terms in the right member of (4.2. 17) ) ; e4) if A C X and for each F in F, A n F E 5 , then A E 5.

A set E in 5 is finite, a-finite or decomposable according as, by abuse of notation, (E, E n 5, It) is totally finite, totally a-finite, or decomposable. Discussions of the relations of the hierarchy to the validity of LRN (and hence the validity of the representation theorems in Section 4.3) can be found in [GeO, Halm, HeS, Loo] .]

4.2 .18 Exercise. If f E L1 (X, p,) , then the set K",(f, O) is a-finite.

4.2 . 19 THEOREM. IF (X, d) IS A METRIC SPACE, (X, aR[K(X)] , p,) IS FINITE, AND EACH x IN X IS THE CENTER OF A a-compact OPEN BALL, THEN (X, aR[K(X)] , p,) IS REGULAR.

PROOF. The set R of regular Borel sets is nonempty since (/) E R. The formulre of set algebra imply that R is a ring.

E If E > 0, R 3 An C An+1 C Un+1 E O(X), and p, (Un \ An) < 2n ' then

00 A �f U An C U Un �f U E O(X) ,

nEN n=2 00 00 00 U \ A C U (Un \ An) , p,(U \ A) � L p, (Un \ An) < L 2: < E.

n=2 n=2 n=2

Section 4.3. LRN and Functional analysis 147

N If U An �f BN, then BN E R. Lebesgue's Monotone Convergence Theo-n=1

rem and the finiteness of (X, aR[K (X)] , I-l) imply that for large N,

E Furthermore, B N contains a compact K N such that I-l (B N \ K N) < 2 ' Hence I-l (A \ KN ) .::;: I-l (A \ B N) + I-l (B N \ K N) < E : A E R.

If E > 0, R 3 Dn :J Dn+1 :J Kn+1 E K(X) , and I-l (Dn \ Kn) < E, then = N

D �f n Dn :J n Kn �f K E K(X) and I-l(D \ K) < E. If n Dn �f EN, nEN n=2 n=1

for large N, EN E R, It (EN \ D) < � , and EN is contained in an open set E UN such that I-l (UN \ EN) < 2 ' Hence

D E R. Thus R is monotone, whence R is a a-ring (v. 2.5. 11 ) . Lebesgue's Monotone Convergence Theorem implies that a a-compact

open ball B (x, r t is regular. If 0 < s < r, then B (x, s t is also regular. If K E K(X) , then for each n in N and each k in K, there is a regular open ball B (k , rkt such that 0 < rk < .!. and so for some finite set {kih<i 0, and for some n, .!. < 15: x tJ- Ur" i.e . , n K = n Un . Since R is a a-ring, K is regular: aR [K(X)] C R. D

nEN

4.3. LRN and Functional analysis

Among the important consequences of LRN are the characterizations of the dual spaces [LP (X, I-l)] ' , and when X is a locally compact Hausdorff space, [Co (X, C)] ' . In particular, the problem raised at the beginning of the Chapter can be addressed. It is no exaggeration to state that modern functional analysis owes its current richness to the role played by LRN in establishing the basic relations among what are now regarded as the classical function spaces.


4.3.1 THEOREM. IF (X, S , p,) IS TOTALLY FINITE ,

1 '::;: p < 00, AND L E [U(X, p,)]' ,

THEN FOR SOME f IN U' (X, p,) , L(g) == Ix gf dp,. THE FUNCTION f IS UNIQUE (MODULO A NULL FUNCTION ) AND I l f l lp' = I I L I I .

[ 4.3.2 Remark. The result above is in one sense a generalization of 3.6.1; the assumption that (X, S, p,) is totally finite limits the generality. On the other hand, extensions to totally a-finite measure spaces are available (v . 4.3.4) . 1

PROOF. At most one such f exists since for any set E in S ,

The heart o f the argument centers on showing that the ([-valued set function � : S 3 E r-+ L (X E) is a complex ,measure, and that � « p,. The

complex conjugate of Radon-Nikodym derivative dp, serves for f . d� The reasons that � is a complex measure and � « p, are: a)

b) L is linear; c) L is continuous (whence for each g in LP (X, p,) ,

Because X is finite, LOO(X, p,) C U(X, p,) . If g E LOO (X, p,) , there is a sequence {sn} nEN of simple functions such that

Sn t g and I l g - sn l loo ..(. 0 (v. Section 2 .1 ) , whence lim I l g - sn l lp = O. Thus, if

n---+oo

then (4.3.3)

(v. 4.2 .13) . The next paragraphs show: f E U' (X, p,) and I f l lp' = I I L I I .

Section 4.3. LRN and Functional analysis 149

If p = 1 and E E 5, then lie I dl-l l .::;: I I L I I . I-l(E) , whence as in the PROOF of LRN, it follows that I I (x) 1 .::;: I IL I I a.e . :

I E L 00 (X, I-l) [= £P' (X, I-l)] .

If 1 < p < 00, then sgn (7) is measurable and sgn (7) 7 = II I (2.4.8) .

If En �f E:; ( II I , n) , n E N, and kn �f xEn ll lpl - 1 , then

I kn lP = xEn ll l pl and kn E LOO(X, I-l) . Hence

1 I l lpl dl-l = r kn · k�- 1 dl-l En Jx = Ix kn l l l = Ix knsgn (7) 7 dl-l = L [knsgn (7)] � I I L I I · l l knsgn (7) l i p � I IL l l l l kn i lp

1 1

and I l kn l lp = (len Il lpl dl-l) P implies (Ix xEn ll lpl dl-l) pr � l i L l i , n E N.

Thus I l h l lpl .::;: I IL I I . Holder 's inequality and (4.3.3) imply I I L I I .::;: 1 1 1 1 1 pl . Thus I I L I I = I l h l lpl , 1 � P < 00. D

4.3.4 Exercise. The conclusion of 4.3.1 holds if (X, 5, I-l) is totally a-finite and 1 '::;: p < 00: [£p (X, I-l)]' = £P' (X, I-l) .

[Hint: If X = U Xn, Xn E 5, and 0 < I-l (Xn) < 00, by abuse nEN . 5 clef 5 clef I ( 5 ) of notatlon, there are n = n Xn, I-ln = I-l Sn ' Xn, n , I-ln , Ba-nach spaces Bn �f LP (Xn , I-ln ) , injections

that identify LP (Xn , I-ln) with closed subspaces of LP (X, I-l), and finally Ln �f L IBn '

For each n there is in Lpl (Xn , I-ln) an hn such that for g in Bn, Ln(g) = r ghn dl-ln and JXn

Each I in LP (X, I-l) may be written uniquely in form 00 00 L IIXn�f L ln' n= 1 n=1


00 n=1

The result 2.3.2 can be extended to

4.3.5 THEOREM. (F. Riesz) IF X IS A LOCALLY COMPACT HAUSDORFF SPACE AND 1 : Co (X, q 3 f r-+ 1(1) E e lS IN [Co (X, q]" THERE IS A REGULAR COMPLEX MEASURE SPACE (X, S,B, p,) SUCH THAT

1(1) = Ix f(x) dp,.

PROOF. If �[1(1)] � 0'(1) , 'S[1(1)] �f (3(1) , then a and (3 are continuous and linear JR.-valued maps, by abuse of language, signed functionals. The argument reduces to showing that a (and similarly (3) is further decomposable into the difference of (nonnegative) DLS functionals to which 2.3.2 applies. There is an echo in what follows of the Hahn decomposition (4. 1 . 10) of a signed measure.

For f in cto(X, JR.) , 0'+ (1) �f sup { la(g) 1 : g E Coo (X, q, Ig l � f } is abbreviated sup l a (g) l . Thus 0'+ (1) 2: 0'(0) = 0 and 0'+ (1) � I l a l l l l f l loo .

Ig l �f If c 2: 0, then

Moreover, if I gi I � fi , i = 1 , 2, then the careful application of the identity sgn (z)z == I z l leads to

0'+ (II ) + 0'+ (h) = sup 10' (gI ) l + sup 10' (g2 ) 1 Ig l l �h I g2 1�h

= sup [ 1 0' (gI ) l + 1 0' (g2) 1 ] 1 9 1 1 Sh 1 92 I Sh

= sup a {sgn [ a (gI )] gl + sgn [ a (g2)] g2} 1 9 1 1 Sh 1 92 1 Sh

.::;: sup 1 0' (gl + g2 ) 1 � 0'+ (h + h) · 1 9 1 1 Sh 1 92 I Sh

If I g l .::;: h + 12 , then 0 � h 1\ I g l �f hI .::;: h , 0 .::;: I g l - hI �f h2 � h Furthermore, since I g l = hI + h2 ,

whence 0'+ (h + h) � 0'+ (h ) + 0'+ (h) · If f E Coo(X,JR.) and

Section 4.4. Product Measures 151

then 0'+ (1) 2: max{O, a(l) } . If 0'- (1) �f 0'+ (1) - 0'(1) , then 0'- (1+ ) 2: 0, a± are continuous, and a = 0'+ - 0'- . Similarly analysis applies to (3. Via the DLS procedure, the functionals a±, (3± engender regular measures (± , 1]± .

If f.-l �f (+ - C + i (1]+ - 1]- ) and f E Coo (X, q and f E Coo (X, q, then 1(1) = r f(x) df.-l, and 1 1 1 1 1 = sup 1 f.-l I (E) . Finally, the II l l oo-dense-� EE� ness of Coo (X, q in Co (X, q applies. D

4.4. Pro duct Measures

To measure spaces (Xi , 5i , f.-li ) , i = 1 , 2, there correspond: a) the space X � Xl X X2 and b) the intersection 5 1 8 52 of all a-rings contained in lfj (Xl X X2) and containing { El x E2 : Ei E 5i , i = 1 , 2 } . There arises the question of how to define a product measure f.-l on 5 1 8 52 so that the equation f.-l (El x E2) = f.-ll (Ed ' f.-l2 (E2 ) is satisfied for all Ei in 5i , i = 1 , 2. The DLS approach to the answer is given below.

[ 4.4 .1 Remark. Alternative derivations can be given by proving whatever is claimed first for simple 5 1 8 52-measurable functions and then, via the approximation and limit theorems of measure/integration, extending the conclusions for wider classes of functions. In such a procedure, the result---a pre-Fubinate measure space (Xl x X2, 5 1 8 52 , f.-ll 8 f.-l2)-can fail to be complete. ]

Associated with (Xi , 5i , f.-li ) are the Banach spaces Ll (Xi , f.-li ) , i = 1 , 2, and the nonnegative linear functionals

Ii : Ll (Xi , f.-li ) 3 f f-t r f df.-li , i = 1 , 2. }xi

If Ap, 1 :::; p :::; P, resp. Bq , 1 :::; q :::; Q, in 51 resp. 52 are sets of finite measure (f.-ld resp. (f.-l2) and apq , 1 :::; p :::; P, 1 :::; q :::; Q, are real numbers, then

P,Q f �f "'" apq X X � Ap Bq

p,q=l is in ffi.x1 XX2 and the set L of all such functions is a function lattice.

P,Q 4.4.2 LEMMA. a) IF "'" apqX X E L, THEN 51 resp. 52 CONTAINS � Ap Bq

p,q=l PAIRWISE DISJOINT SETS Eu , 1 :::; u :::; U, resp. Fv, 1 :::; v :::; V, AND ffi. CON-TAINS NUMBERS auv , 1 :::; u :::; U, 1 :::; v :::; V, SUCH THAT

P,Q u,v

L apqXApXBq = L auvXEuXFv ' p,q=l u,v=l


b) FURTHERMORE,

P,Q U,V

L apqh (XAJ 12 (XBJ = L O'uvh (XEJ h (XFJ · p,q=l u,v=l

PROOF. a) A detailed computational argument establishes the validity of the assertions in the LEMMA. Not only is the argument tedious, it is not really informative and adds little to the understanding of the underlying structure shown in the visible geometry. The basic reasoning is depicted in Figure 4.4 .1 . The interested reader might wish to provide a formal argument that translates the geometry into the unavoidable prolixity.

b) The linearity of the functionals h and h implies the result. D

P,Q L clef 4.4.3 Exercise. If apqX X = f E L, then Ap Bq

p,q=l

is well-defined, i .e . , 1(1) is independent of the representation of f.

UI x V3 U2 x V3 U3 x V3

UI x V2 U2 x V2 U3 x V2

UI x VI U2 x VI U3 x VI

L---�------�----------�--------�---- XI U2

Figure 4.4. 1 .


4.4.4 DEFINITION . WHEN A C Xl X X2 AND X E Xl , THE x-section Ax OF A IS { X2 : (x, X2) E A } . WHEN f E ffi.X1 XX2 THE x-section fx OF f IS X2 : 3 X2 r-+ f (x, X2) . SIMILAR FORMULATIONS APPLY FOR Y IN X2 , Ay , AND fy .

4.4.5 LEMMA . THE MAP I IS A DLS FUNCTIONAL. PROOF. The nonnegativity and linearity of I flow from its definition. FurP,Q thermore, if f �f " apqX X E L, then for each (fixed) x in Xl , � Ap Bq p ,q= l

P,Q fx = L apqX A (x)XB E LI (X2' f.-l2) , p q p,q= l

P,Q 12 (fx ) = L apqXAp (x)h (XBJ E Ll (Xl , f.-ld ·

p ,q=l Similar formulre apply for fy . The definitions of the functions imply the fundamental equality: 1(f) = II [12 (fx )] = h [h (fy ) ] .

Because (fn )x ..!. 0, h [(fn U ..!. O. Furthermore,

Ll (Xl , f.-ld 3 h [(fn)x ] ..!. 0 , whence h {h [(fnU} = I (fn ) ..!. o . D

Since I in 4.4.5 is a DLS functional, in accordance with the developments in Chapter 2, I engenders a complete measure space (X, 5, f.-l) , the Fubinate of (Xl , 5 1 , f.-ld and (X2' 52 , f.-l2 ) :

When the number of ingredient factor spaces is two or more, the general vocabulary and notation for dealing with product measure spaces are those given in

4.4.6 DEFINITION. a) FOR A SEQUENCE { (Xn, 5n , f.-ln ) }nEN OF MEASURE SPACES, : (Xl x X2, 5 1 X 52, f.-ll x f.-l2) IS THE Fubinate OF (Xl , 51 , f.-ld AND (X2 ' 52 , f.-l2) ' b) FOR K GREATER THAN 2 ,

I S THE Fubinate OF

(XK-l XK-l XK-l ) Xk, 5k , f.-lk k=l k=l k= l


clef XK FOR Y = Xk , THE INTERSECTION OF ALL a-RINGS CONTAINED IN k=l !fj(Y) AND CONTAINING { X�=l Ek : Ek E Sk , 1 ::; k ::; K } IS 6Sk. k=l

K K K

ON OSk, Xk=lf.-lk I S A MEASURE DENOTED Of.-lk AND THERE EMERGES k=l k=l THE pre-Fubinate MEASURE SPACE (X�=l Xk, g Sk, g f.-lk) . WHEN

Xi == X, EACH Si == 5, AND EACH f.-li == f.-l, THE FORMULJE

PROVIDE THE NOTATIONS FOR THE K-FOLD FUBINATE OF (X, S , f.-l) WITH ITSELF AND THE K-FOLD PRE-FuBINATE OF (X, 5, f.-l) WITH ITSELF.

K K

4.4.7 Exercise. a) If K 2: 2, then Xk=l Sk :J OSk . b) If Ek E Sk , then k=l (X�=lf.-lk) (X�=l Ek) = fl f.-ldEk) '

From 4.5.10, 4.5.16, and 4.5.18 below it follows that m some m-stances :J in 4.4.7a) is = and in others :J is -;;. .

Fubinate measure spaces arise from DLS functionals and are automatically complete. Some pre-Fubinate measure spaces can fail to be complete. The procedure described in 4.4.1 leads to pre-Fubinate measure spaces.

Their completions are the Fubinate measure spaces. One advantage of the DLS approach to product measures is the fact that it immediately produces Fubinate measure spaces.

4.4.8 Exercise. a) If (X, 5 , f.-l) is a measure space and 5 contains a ring R of sets of finite measure, then aR(R) is a-finite. b) The measure space (X, 5, f.-l) engendered by I in 4.4.5 is a-finite.

[Hint: As in 2.2.40, for a) the relevant sets are Eo, the set of all countable unions of elements of R, and for each ordinal number a in (0, Q) , the set En of unions of count ably many set differences drawn from U E-y .J

-y<n


The machinery of Chapter 2 applies in the current situation and leads to

4.4.9 THEOREM. (Fubini) FOR MEASURE SPACES (Xl , S; , IL; ) , i = 1 , 2, AND THE PRE-FuBINATE MEASURE SPACE

Ix E LI (X2 ' 1L2 ) resp. Iy E LI (Xl , ILl ) , x r-+ r Ix (Y) dIL2 (Y) E 51 resp. Y r-+ r Iy (x) dILl (x) E 52 ,

iX2 ix!

Ix I dlL = Ix! (lx

2 Ix d1L2) dILl = 1x2

(Ix!

Iy dILl) d1L2 . PROOF. Since L is l i l l I -dense in . Ll (X, IL) (v. 2.2 .31 ) , for a sequence {fn}nEN contained in L, lim I l ln - 1 1 1 1 = 0, whence n---+=

r I dlL = lim I (f n) . ix n---+=

Via, as needed, passage to a subsequence, recourse to the lattice operations 1\ and V, and the replacement of I by a g such that g == I (g and I are indistinguishable in L 1 (X, IL) ) , 0 � In t I may be assumed. For each x in Xl resp. each Y in X2, 0 � (fn )x resp. 0 � (fn )y and

(4 .4.10)

resp. (4.4 . 1 1 )

Owing to the form of each In,

Ix In dlL = Ix! (lx

2 (fn )x d1L2) dILl

= 1x2 (Ix

!

(fn )y dILl ) d1L2 . (4 .4.12)

Lebesgue's Monotone Convergence Theorem (v. 2 .1 .9f) ) applies. D

4.4.13 Exercise. In the context of Fubini's Theorem (4.4.9), if


then for x in Xl resp. y in X2, Ix E LI (X2, f.-l2) resp. Iy E LI (Xl , f.-ld and (4.4.12) obtains.

[Hint: In L l (X, f.-l) there are nonnegative functions 1; , 1 :::; i :::; 4 , such that I = II - h + i (13 - 14) ']

4.4.14 Exercise. For (Xl x X2, 5 1 8 52 , f.-ll 8 f.-l2) and a nonnegative function I in L l (Xl X X2, f.-ll X f.-l2 ) , for almost every x in Xl ,

Ix E LI (X2, f.-l2 ) ' J Ix d (f.-l2) E 51 , x2

Ixl XX2 = Ixl (lx2

Ix df.-l2) '

Similar formul� obtain for almost every y in X2 • Fubini's Theorem and its elaborations describe the circumstances in

which an integral of an l over a product space may be calculated by iterated integration of the sections Ix over the corresponding factor spaces. Although no conditions, e.g. , a-finiteness, are imposed on the measure spaces (Xi, 5i , f.-li ) , i = 1 , 2, the PROOF deals only with an integrable function I, and automatically K",(f, 0) is a-finite.

The next result describes circumstances in which the existence of an iterated integral of the sections of l over the corresponding factor spaces implies the existence of the integral of l over the product space (in which case the two calculations yield the same result) .

4.4.15 THEOREM. (Tonelli) FOR A TOTALLY FINITE MEASURE SPACE

(Xl X X2 , 51 X 52 , f.-ll X f.-l2) ' IF I IN ffi.X1 xX2 IS NONNEGATIVE AND 5 1 X 52-MEASURABLE AND EITHER OF

Ixl (lx2 Ix df.-l2) df.-ll < 00,

1x2 (lx l Iy df.-ll) df.-l2 < 00,

OBTAINS , SO DOES THE OTHER, THEIR VALUES ARE EQUAL,

AND

Ixl xX2 I d (f.-ll X f.-l2) = Ixl (lx2

Ix df.-l2) df.-ll

= 1x2 (lx l Iy df.-ll) df.-l2 ·

( 4.4.16)

(4.4 . 17)

(4.4 . 18)


PROOF. Since I is measurable, there is a sequence {In } nEN of simple functions such that In t I. Each of the following equalities is validated by Lebesgue's Monotone Convergence Theorem (v. 2 .1 . 19f) ) or by the definition of integration of simple functions:

r i d (f.-l1 x f.-l2) = lim r In d (f.-l1 X f.-l2) }X1 XX2 n-too }X1 XX2 = }�+� 1x2 [lxl (fn )y df.-l1 ] df.-l2

= 1x2 [J�� Ixl (fn )y df.-l1] df.-l2

= 1x2 [/XI nl�� (fn )y df.-l1] df.-l2

= 1x2 [/XI Iy df.-l1] df.-l2 .

Hence, if the last (iterated) integral is finite so is the (double) integral and Fubini's Theorem implies the conclusions of the theorem. D

4.4.19 Exercise. Tonelli's Theorem is valid if: a) each measure space (Xi , 5i , f.-li ) , i = 1 , 2, is a-finite; b) I is 5 1 x 52-measurable.

4.4.20 Exercise. a) If E E 5 � n, there is a Borel set F such that

(E \ F) U (F \ E) (= Ef:l.F)

is a null set (A x n) . b) If I is 5 � n - measurable, there is a Borel measurable g such that 1 == g. c) If h : ffi.2

r-+ ffi. is Borel measurable and I : ffi. r-+ ffi. is Borel measurable, then I 0 h is Borel measurable.

[Hint: For c) use b) . ]

4.4.21 Exercise. If {J, g} C L1 (ffi., A ) , then

h : ffi.2 3 (x, y) r-+ I(x - y)g(y)

l I(x - y)g(y) dy �f 1 * g(x) ,

the convolution of I and g, exists a.e. , and


[Hint: The functions f and g may be assumed to be Borel measurable; JR.2 3 (x, y) f-t x - Y E JR. is continuous. Fubini's Theorem applies to r f(x - y) (g(y) d).. X 2 (x , y) .]

JK{2

[ 4.4.22 Note. If # (A) 2: No, an indexed family ((X)" S)" 1-l), ) }),EA of measure spaces, X X), has a standard meaning. In construct-),E), ing a measure I-l related to the set {1-l),hEA and on some a-ring related to {S)'hEA awkwardness is avoided if the conditions: a) for each >.. , X), E S), and b) I-l)' (X), ) = 1 are imposed.

The customary approach is to form, for each finite subset

the rectangle R �f E)'l X . . . x E), x X X)" then the algebra n ),$a

F, and the a-algebra A generated by all such rectangles. For each n

R, I-l( R) �f II I-l),k (E)'k ) ' The set function I-l is extensible to a k= l

measure on A and this measure behaves properly with respect to the set {1-l),hEA of given measures.

Details of this procedure and of the associated Fubinoid theorems can be found in [HeS] .]

4.5. Nonmeasurable Sets

The existence of nonmeasurable sets in the context of a measure space (X, S, I-l) can be established trivially in some instances and in others, only appeal to sophisticated aspects of set theory permits a satisfactory resolution.

4.5.1 Example. a) If X i- (/) and R �f {(/)}, for I-l : R 3 E f-t 0, (X, R, I-l) is a complete measure space and every nonempty subset of X is nonmeasurable. b) The set C consisting of (/) and all finite or countable subsets of JR. is a a-ring and v : C 3 E f-t #(E) is counting measure. If X �f JR., the measure space (X, C, v) is complete, and every non countable subset of X, e.g. , JR., is nonmeasurable. (Note that if E E R resp. E E C and M e X, then M n E E R resp. M n E E C. )

By contrast , the following discussion demonstrates that for the complete measure space (JR., S)" )..) , there is a set M such that if )"(E) > 0, then

Section 4.5. Nonmeasurable Sets 159

M n E tJ- 5), . The result is applied to the study of the completeness of product measure spaces that arise from complete measure spaces. 4.5.2 Example. The complete a-algebra 5 generated by the set of all arcs A:>,(3 �f { eiO : 0 ':::; a < (J < (3 .:::; 27r } in 'lI' may be endowed with the measure 7 (v. 3.5.47) such that 7 (Aa,(3 ) �f (3 - a. If (J E JR., then eiO Aa,(3 E 5 and 7 (e'O Aa,(3) = (3 - a: ('lI', 5 , 7) is translation-invariant.

In the group 'lI' there is the subgroup G �f { eiO : (J E Q } . The Axiom of Choice implies the existence in 'lI' of a set 5 meeting each coset of G in precisely one element . Thus { g5 : 9 E G } consists of count ably many pairwise disjoint sets such that U g5 = 'lI'. If 5 E 5 and 7(5) = a, then gEG for 9 in G, 7(g5) = a: for each n in N, 7('lI') 2: na, and so a = O. Since 'lI' = U g5, it follows that 7('lI') = 0, a contradiction. Thus 5 C!fj('lI') . gEG 'f=-The map exp : JR. 3 x r-+ eix E 'lI' (v. 2.3.13) permits a transfer of the discussion above from ('lI', 5, 7) to (JR., 5)" A) . The conclusions reached are paraphrased loosely by the statement: 'lI' and JR. contain nonmeasurable sets. 4.5.3 Exercise. a) For disjoint sets El and E2 ,

(4.5.4 )

b) for each F in 5 and E in H (5 ) , p,* (F n E) + p,* (F \ E) = p,(F) ; c) if E E H (5) and p,* (E) < 00, then E E 5 iff p,* (E ) = p,* (E) .

[Hint: a) For the first inequality in (4.5.4) there are sequences {An} nEN resp. {Bn} nEN contained in 5 and such that

An C El and P, (An ) t p,* (Ed , Bn :J E2 and p, (Bn ) ..(. p,* (E2 ) ;

for the second inequality, there is an 5-sequence {en} nEN such en :J en+l and p, (en ) ..(. p,* (Ell:!E2) . Then p, (An n en) t p,* (Ed and P, (Bn n en) ..(. p,* (E - 2) . For b) , a) applies.]

4.5.5 Example. If a is an irrational real number, � �f e27ria , A �f { C : n E Z } , and B �f { en : n E Z } ,

then: a) B is a subgroup of index 2 in A; b) B n �B = (/), and A = Bl:!�B; c) because a is irrational, both A and B are (countably) infinite dense subgroups of the compact group 'lI'. Zorn's Lemma implies there is a set 5 consisting of exactly one element of each 'lI'-coset of A. For M � 5B, if M M-l n �B i- (/), i.e. , if


since '][' is abelian, S 1 S;- 1 E �B C A. Hence, owing to the nature of 5, S 1 = 82 . Thus X1X;- 1 = b1 b;- 1 E B, i .e . , X1 X;- 1 E �B n B = 0, a contradiction:

MM�1 n �B = 0.

If L is a measurable subset of M and T(L) > 0, then M M�1 :J LL �1 , which contains a ,][,-neighborhood of 1 (v. 2.3.9) and thus an element of the dense set �B, a contradiction. It follows that the inner measure of M, i.e. , the supremum of the measures of all measurable subsets of M, is zero: T* (M) = O. Because T is translation-invariant , T* (�M) = O.

For x in '][' there is in 5 an s such that XS� 1 � a E A. If x tJ. M, then a tJ. B: for some b in B, x = s�b E 5�B = �M. Thus '][' \ M c �M, and so T* ('][' \ M) = O. For each measurable set P,

T* (P n M) + T* (P \ M) = T(P) ,

(v. 4.5.3) , whence T* (P n M) = T(P), in particular,

T* (M) = 1 > 0 = T* (M) .

The set (}� 1 (M) � M in JR. has properties analogous to those of M .

• The set M is nonmeasurable, ),* (M) = 0, and ), * ( M) = 00 .

• The set M is thick and for every measurable subset P of JR.,

),* (P n M) = 0 while ),* (P n M) = ),(P)

(whence if )'(E) > 0, then M n E is nonmeasurable) .

[ 4.5.6 Note. The use of Zorn's Lemma or one of its equivalents is unavoidable in the proof that !fj(JR.) \ 5), i- 0. Solovay [Sol] shows that adjoining the axiom:

Every subset of JR. is Lebesgue measurable. to ZF, the Zermelo-Fraenkel system of axioms of set theory, yields a system of axioms no less consistent than ZF itself. ]

4.5.7 Exercise. For (X, 5, f..l) , if f..l* resp. f..l* are the associated �uter and inner measures induced by f..l on H(5) , then a f..l* -a-finite E is in 5 iff

4.5.8 Exercise. a) If f : JR. r-+ JR. is (Lebesgue) measurable and p is a polynomial function, then p 0 f is measurable. b) If g E C(JR., JR.) , then g 0 f is measurable. c) For the Cantor set Co , the corresponding Cantor function

Section 4.5. Nonmeasurable Sets 161

cPo and the function '1jJo (v. 2 .2 .40 and 2 .2 .41 ) are continuous and: c1) '1jJ (10,20 ) = ( 1

52 ' 1�) �f J contains a set that is Lebesgue nonmeasurable;

c2) '1jJ-l (5) �f g(5) is Lebesgue measurable; c3) h �f X,p- l (S) is Lebesgue measurable; c4) h o g is not Lebesgue measurable.

In sum: A continuous function composed with a measurable function is measurable; a measurable function composed with a continuous function can fail to be measurable.

[ 4.5.9 Note. For measure spaces (Xi , 5i , f.-li ) , i = 1 , 2 , the Fubinate (Xl x X2 , 5 1 X 52 , f.-ll X f.-l2 ) derived in Fubini's Theorem is automatically complete (v. 2 . 1 .42d) ) and 51 x 52 :J 5 1 8 52 . There follow instances (4.5.10) where :J is = and others (4.5.18) where :J is �.l

4.5.10 Example. If 5i �f R or 5i �f C, i = 1 , 2 , in 4.5.1 , then

4.5 .11 THEOREM. FOR FINITE MEASURE SPACES (Xi , 5i , f.-li ) , i = 1 , 2 , THE FUBlNATE (Xl X X2, 5 1 X 52 , f.-ll x f.-l2 ) IN 4.4.9 IS THE COMPLETION OF THE PRE-FuBINATE (Xl X X2, 5 1 8 52 , f.-ll 8 f.-l2 ) ' PROOF. In the context and notation of the general DLS construction of Chapter 2, L is l i l l I -dense in Ll (X, f.-l) . Hence, if E E D and f.-l(E) < 00, then X E is the I I I l l -limit of a sequence {f n} nEN in L. In the current context (cf. 4.4.2 and 4.4.5), each In is a simple function with respect to 51 8 52 . Via passage to a subsequence as needed, it may be assumed that lim In �f I exists a.e., and with respect to f.-ll x f.-l2 , I � X . If each In n---+= E

is replaced by clef { 1 gn = o

if In i- 0 otherwise '

then {gn} nEN C L, lim gn �f g exists a.e. , and with respect to f.-ll x f.-l2 , n---+=

If Fn �f { (x, y)

1 · F clef F Inl n = n---+=


exists and is in 51 x 52 , and f..l1 x f..l2 (Ft:J.E ) = O. D

4.5.12 Exercise. The conclusion to 4.5.11 remains valid if the measure spaces are a-finite.

4.5.13 COROLLARY. THE FUBINATE OF (JR.k , sTk , ),0k) AND (JR.1 , ST1 , ),01 ) IS THE COMPLETION OF THE PRE-FuBINATE (JR.k+l , S�(k+l) , ),0(k+l) ) .

4.5.14 Exercise. If L �f Coo (JR.n , JR.) and I is n-fold iterated Riemann integral: I : Coo (JR.n , JR.) 3 f r-+ l (l C · · (l f dXn) . . . ) dX2) dX1 , then I is a DLS functional. The result of applying the DLS procedure in the context of L and I is a measure space that is the n-fold Fubinate of (JR., 5)" ),) with itself.

[ 4.5.15 Note. The n-fold Fubinate of (JR., 5), , ),) with itself is denoted (JR.n , S),n ' ),n) ; ),n is n-dimensional Lebesgue measure .]

4.5.16 Example. For the set M of 4.5.5,

5 �f {O} x M C {O} x JR. E S� 2

and ),X 2 ( {O} x JR.) = 0: 5 is a subset of a null set (),X 2) . If 5 E S�2 , then 50 = M E 5)" a contradiction: (JR.2 , ST2 , ),02) is not complete; (JR.2 , S�2 ' ),2) is complete.

4.5. 17 THEOREM. IF {k, l} e N, THE COMPLETION OF

(TTllk+l 5 \ ) S0(k+l) - 5 IS m. , ),k+l ' Ak+l : ), - ),k+l ' PROOF . Since JR.n is totally a-finite with respect to both ),n and ),xn , the results in 2 .5.13 and 2 .5.14 apply. D

4.5.18 COROLLARY. S� (k+l) �S�(k+l) .

[ 4.5.19 Note. When n > 1 and confusion is unlikely, the notation (JR.n , 5), , )') replaces the unwieldy (JR.n , S)'n ' ),n ) and the associated usages are simplified.]

4.5.20 Exercise. The measure space (JR.n , 5), , )') is translation-invariant and rotation-invariant, i .e . , for T a translation or a rotation of JR.n , if E E 5)" then T(E ) E 5), and )' [T(E)] = ),(E) .

Section 4.5. Nonmeasurable Sets

[Hint: The a-ring generated either by the set of all open balls or the set of all open rectangles

contains 5 � n .J

163

4.5.21 THEOREM. IF (lR.n , 5),n ' f.-l) IS A MEASURE SPACE SUCH THAT EACH HALF-OPEN INTERVAL xn [ak , bk ) IS OF FINITE f.-l-MEASURE AND f.-l IS k= l TRANSLATION-INVARIANT, FOR SOME NONNEGATIVE CONSTANT p,

PROOF. If f.-l ( [0, l )n ) �f p, then, since [o, l )n is the union of 2kn pairwise disjoint half-open subintervals Ij , each the translate of any other,

Thus f.-l( E)) = p for every half-open interval E and hence the equality An (E obtains for every E in the a-algebra generated by the set of half-open

��. D As remarked earlier, 4.4.9 imposes no restriction on the underlying

measure spaces. Since f E Ll (X, f.-l) , E,,,, (f, 0) is a-finite, i .e. , the integration with respect to f.-l is performed over a a-finite set. However, if only 5-measurability for f is assumed, K",(f, O) can fail to be a-finite and the conclusion of Fubini's Theorem can fail.

4.5.22 Example. If Xl = X2 = JR., 51 = 5)' , 52 = !fj(JR.) , f.-ll = ).., and f.-l2 is counting measure v , then E �f { (x , y) : x = y } E 51 X 52 �f 5, the characteristic function f �f X E of the diagonal E �f { (x , y) : x = y } is 5-measurable, and

In this instance, K",(f, O) is not a-finite. Indeed, f is not integrable with respect to f.-ll x f.-l2 .


4.6. Differentiation

The symbol �: that appears in LRN is reminiscent of the classical symbol dy d . ( [ ] clef lX - use m elementary calculus. If f E e 0 , 1 , q , y = f(t) dt, and dx 0

� : S), 3 E r-+ Ie f(t) dx � �(E) , then � « >. and

dy dx lim >.( [x ,x+h] ) -+O h>,o

J:+h f(t) dt h

� ( [x , x + h] ) lim >.([x ,x+h])-+O >.( [x x + h] h>,o ' d�

= - = f(x). d>'

(4 .6 . 1)

(4.6.2)

(4.6.3)

The display above suggests that for the measure space (JR., S)" >.) and a complex measure space (JR., S), , p,) such that p, « >.,

lim p,(E) . >' (E )-+O >.(E) >'( E)>'O (4.6.4)

However, the left member of (4.6.4) is a point function, i .e . , �� E ClR , while there is no reference to any point of JR. in the right member. The resolution of this difficulty is addressed in the following paragraphs. A particular consequence of the development is a useful form of the Fundamental Theorem of Calculus (FTC) (v. 4.6.15, 4.6.16, and 4.6.33) :

If f E L l (JR., >.) and F(x) �f lx f d>', then F' exists a.e. (>.) and F' � f. In the wider context of Lebesgue measure and integration, FTC is a

corollary to several more general results that will emerge as the discussion proceeds.

[ 4.6.5 Note. Most of the next conclusions are true almost everywhere ( a.e. ) , not necessarily everywhere. The calculations involve measures of unions of sets. Because measures are count ably additive, the arguments are eased if the constituents of the unions are pairwise disjoint. The context is a complex measure space (JR.n , S)" p,) . The Jordan decomposition of p, and LRN permit the assumption that p, is nonnegative and totally finite, whence regular (v. 4.2. 19) . Thus attention can be focussed on open sets, since the measure of an

Section 4.6. Differentiation

arbitrary measurable set E can be approximated by the measures of open sets U containing E. The following constructions allow much of the argument to be confined to special collections of half-open cubes Q ' " and associated open cubes C. . . . These are used as well in a different context in Section 7.1 .]

For m in N and k �f (kl , . . . , kn ) in Zn , clef { clef k; ki + 1 . } Qk,m = x : X = (Xl , . . . , xn ) , 2m :::; Xi < � , 1 :::; l :::; n

165

clef ( kl kn ) is the half-open cube vertexed at (k, m) = 2m ' . . . , 2m . For each k and

m, >. [8 (Qk,m)] = 0, and thus >.(N) �f >. [ U 8 (Qk,m)] = O. kEzn ,mEN

Since (lR.n , S)" >.) is complete, every subset S of N is in S), and >.( S) = O. All null sets S emerge as ignorable in the following presentation. Unless the contrary is stated, null set, a .e . , measure zero, =, etc. , are to be interpreted as referring to Lebesgue measure >.. The open cube Qk,m [= Qk,m \ 8 (Qk,m )] is denoted Ck,m , while the

closed cube Qk,m is denoted Rk,m ' For the remainder of this section, ffi.n will be regarded as endowed with its customary (product) topology.

If A and B are two cubes each of which may be half-open (a Q . . . ) or open (a C . . . ) and A n B i- 0, then one is a subset of the other: A and B are disjoint or nestle. Hence any union of such cubes is the union of pairwise disjoint cubes.

• If m E N, then ffi.n = U Qk m ' kEZn ' • If X E ffi.n , then x is in some unique QkJ , I (.x) , in some unique Qk2 , 2 (x) ,

. . . , i .e . , there is a decreasing sequence

(4.6.6)

of uniquely determined half-open cubes such that x = n Qk=,m(X) .

• If x E ffi.n \ N, there is a unique decreasing sequence

such that x = n Ck= ,m (X) . mEN

mEN

(4.6.7)


4.6.8 LEMMA. IF U IS AN OPEN SUBSET OF JR.n , THEN: a) U IS THE UNION OF COUNTABLY MANY PAIRWISE DISJOINT HALF-OPEN CUBES Q . . . ; b) FOR SOME NULL SUBSET S OF N, U \ S IS THE UNION OF COUNTABLY MANY PAIRWISE DISJOINT OPEN CUBES C . . . .

PROOF . a) Since U is open, if x E U, then some B(x, rt is contained in U. For large m, some Qk,m is contained in B(x, rt. Since the set of all half-open cubes is countable, U is the union of a countable set of half-open cubes. Since any two such are disjoint or nestle, if they nestle, the smaller of the two may be discarded. The open set U remains the union of those not discarded.

b) If, as in the argument for a) , U = U Q . . . , then

and U \ S = U C . . . . D

4.6.9 Exercise. If U is an open subset of JR., then U = (/) or U is the union of a unique finite or countable set of pairwise disjoint open intervals .

4.6.10 DEFINITION. FOR A COMPLEX MEASURE SPACE (JR.n , SA , f.-l) , AND x IN JR.n , WHEN {Ck= ,m (x) }mEN ' IS THE SEQUENCE IN (4 .6 .7 ) ,

IF x E N OTHERWISE '

. f.-l [Ck= ,m(X)] WHEN x tJ- N AND hm \ [ ( )] EXISTS, IT IS THE derivative Df.-l(x) m---+= /\ Ck m X at x of f.-l with respect to A.

= ,

4.6 .11 LEMMA . a) IF r E JR. AND U �f { x : M (x) > r } , THEN U IS OPEN, i.e. , M IS lsc. b) IF r > 0, THEN A(U) :::; I f.-l l (JR.n ) .

r PROOF. a) If r :::; 0, then U = JR.n . If r > 0 and x E U, then for· some m,

I f.-l l [Ck=,m(X)] > rA [Ck= ,m(X)] . If Y E Ck= ,m (x) , Ck= ,m(Y) = Ck= ,m(X) , whence M (y) > r, i .e . , y E U: Ck= ,m(x) C U. Since Ckm ,m(x) is open, U is open.

b) Each x in U is in some open cube C . . . (x) contained in U and such that I�I [�::(��] > r. If {Cp}PEN is an enumeration of the cubes C . . . (x) as x varies over U, there is a subsequence {Cpq } qEN defined inductively as follows:

Section 4.6. Differentiation 167

.' CP1 = C1 ;

.' if CP1 , • • • , Cpq have been defined, are pairwise disjoint, and

u = u q Cp j=1 J

the induction stops. If U i- U :=1 Cpj , Pq+ 1 is the least P greater than

Pq and such that Cp is disjoint from U q Cp . • J= 1 J

Since two cubes C . . . are disjoint or nestle, U = U Cpq and for each q, q IIL I/Cpq/ > r. Thus )'(U) = "' ), (Cp ) < '" I IL I (CpJ ::::: I ILI (l�n ) . D

), Cp � q � r r q q q

4.6.12 Exercise. If f E L1 (l�n , IL) , IL : 5 3 E r-+ IL(E) �f Ie f d)" and

. 1 1 hm ), [C ( )] I f (y) - f(x) 1 d)'(y) = 0 , m---+= k ,m X Ck.m (x)

then DIL(X) = f(x) (v. 4.6. 15) . 4.6.13 Exercise. I f g E C(l�n , q, then for each x,

lim ), [C 1 ( ) ] 1 I g(y) - g(x) 1 d),(y) = O. m---+= k,m X Ck,m (X)

4.6.14 Exercise. If f E L1 (l�n , ),) and r > 0, then

), ( { x : I f(x) l > r } ) ::::: ll1.lh, r sup 1 1 I f (y) 1 d)'(y) > r }) ::::: ll1.lh.

1:$m<= ), [Ckm ,m(X)] Ckm ,m(X) r

4.6.15 THEOREM. IF f E L1 (l�n , IL) , THEN

lim ), [ 1 ( )] 1 I f (y) - f(x) 1 d)'(y) � O. m---+= Ck,m X Ck,m (X)

[ 4.6.16 Remark. The set Lf of x where

lim ), [C 1 ( )] 1 I f(y) - f(x) 1 d)'(y) = 0 m---+= k,m X Ck.m(x)


is the Lebesgue set of f. In view of 4.6.12, 4.6.15 implies that for the measure f.-l : S), 3 E r-+ Ie f(y) d)..(y) , the Radon-Nikodym

d ' . df.-l . f ] envatlve d)" IS a.e.

PROOF. The map f.-lx : S), 3 E r-+ le l f(Y) - f(x) 1 d)"(y) is a measure. The assertion to be proved is that Df.-lx (x) exists a.e. and that the result is zero a.e. The conclusion is equivalent to the statement that if r > 0 the set of

. t . N d h -1' f.-lx [Ck= m(X)] . 11 t pom s x not m an w . ere 1m [ ' ( )] > r IS a nu se . m---+= ).. Ck= ,m X Since Coo (lR,n , q is a dense subset of L 1 (lR,n , )..) if E > 0 there is in

Coo (lR,n , q a g such that I l f - g i l l �f I l h l l l < E. Furthermore,

However,

).. [Ck:,m (X)] ik= .= (X) I f(Y) - f(x) 1 d)"(y)

� ).. [Ck:,m (X)] ik= '=(X) I h(y) 1 d)"(y) + I h(x) 1 + ).. [Ck:,m (X)] ik= .= (X) Ig(y) - g(x) 1 d)"(y)

clef II = Im + IIm + I m'

lim IIm > -- r } m---+= 3

Owing to 4.6.13, the last of the three summands above is empty; owing to 4.6.11 and 4.6. 14, the measure of each of the first two summands does 3E not exceed - . D

r

4.6.17 Exercise. a) For some x-free constant Kn, if

x E lR,n \ N and r > 0,

there is a Ck,m(X) containing B(x, rt and such that


b) For some x-free constant Ln,and each Ck,m(X) , there is a B(x, rt containing Ck, m (x) and such that

1 1 >. [B (x, r)O ] > Ln >. [Ck,m (X)] '

4.6.18 Exercise. If f E L l (lR.n , >.) and r ..(. 0, for all x off a null set,

. 1 1 hp >. [B( )0 ] I f(y) - f(x) 1 d>.(y) = 0, r+O x, r B (x,r) O

. 1 1 f(x) = hm >. [B( )0 ] f(y) d>.(y) .

rto x, r B (x,r) O

[Hint: For Ck,m(X) and B(x, rt as in 4.6.17a)

K" >. (B( 1 )0 ) r I f (y) - f(x) 1 d>.(y) x, r } B (x,r) o

� >. [C 1 (x)] 1 I f (y) - f(x) 1 d>.(y) . ] k,m Ck,m (X)

[ 4.6. 19 Remark. In [Rud] nicely shrinking sequences and in [Sak] sequences with parameters of regularity are introduced to define alternative versions of Df.-l. Since two sets A and B underlying those treatments can fail to be disjoint or to nestle, covering theorems related to Vitali 's Covering Theorem are used to cope with this situation. The cubes C . . . require no appeal to such deVIces. The burden of 4.6. 17b) is that for each x in not in N there is a nicely shrinking sequence {Ck,m(X) } mEN ' Similarly, 4.6.17a) says that for each x not in N there is a sequence {B (x, r m) ° } mEN that is nicely shrinking relative to some sequence in Q.]

There remains a collection of results dealing with three classes of functions specified in the terms and symbols of

4.6.20 DEFINITION . WHEN P IS A RIEMANN PARTITION (cf. 2.2.50) OF [a, b] AND h E ClR :

n a) THE P-variation OF h IS varh (P) �f L I h (Xk ) - h (Xk-d l ; THE total k=2

variation OF h ON [a, b] IS

P a Riemann partition of [a, b] } ;


b) WHEN varh ( [a, b] ) < 00, h IS OF bounded variation on [a, b] :

h E BV( [a, b] ) ;

WHEN sup varh ( [a , b] ) < 00, h I S O F bounded variation: -oo<a<b<oo

h E BV;

c) h IN ffi.[a,b] IS monotonely increasing ON [a , b] (h E MON( [a , b] ) ) IFF

{a � x < y � b} ::::} {h(x) � h(y) } ;

d) WHEN 9 E [0, (0) [a ,b] AND h E MON([a, b] ) , THE upper Riemann-Stieltjes sum AND resp. lower Riemann-Stieltjes sum ASSOCIATED WITH THE TRIPLE {g, h, P} IS

resp.

n Sp(g, h) �f L sup g(x) . [h (Xk ) - h (xk-d]

k=2 Xk- l :SX:SXk

n sp (g, h) �f L inf g(x) . [h (Xk ) - h (xk-d] . Xk_ l <X<Xk k=2 - -

THE Riemann-Stieltjes integral lb 9 dh OF 9 WITH RESPECT TO h

EXISTS WHEN

inf { Sp(g, h) : P a Riemann partition of [a, b] } = sup { sp (g, h) : P a Riemann partition of [a , b] }

AND lb g dh IS THE COMMON VALUE OF THE TWO MEMBERS ABOVE;

dl) WHEN 9 E ffi.[a,b] , lb

9 dh EXISTS IFF lb g+ dh AND lb

g- dh EX

IST AND lb g dh = lb

g+ dh - lb g- dh;

d2) WHEN 9 E ([[a,b] , THE RIEMANN-STIELT JES INTEGRAL lb 9 dh

b EXISTS IFF THE RIEMANN-STIELTJES INTEGRALS 1 �(g) dh AND

lb 'S(g) dh EXIST; FURTHERMORE


e) h IS absolutely continuous on [a, b] (h E AC( [a, b] ) ) IFF FOR EACH POSITIVE E THERE IS A POSITIVE r5 SUCH THAT WHENEVER

THEN

WHEN h E ClR AND h IS IN AC([a , b] ) FOR EVERY [a, b] , THEN h IS absolutely continuous (h E AC) .

4.6.21 Exercise. a) The Cantor function cPo: (2.2.41) is in MON([O, 1] ) and in BV( [O, 1] ) but not in AC([O, 1] ) . b) If f E L l ([a , b] , >.) and, for x in [a , b] , F(x) �f lx

f(y) d>.(y) , then F is in AC([a, b] ) . c) The map

g : [0, 1] 3 x r-+ { XC sin � if x E (0, 1] o if x = 0

is in BV([O, 1] ) if c > 1 but is not in BV( [O, 1] ) if c = l .

4.6.22 Exercise. MON([a, b] ) U AC([a, b] ) C BV([a, b] ) . 4.6.23 Exercise. a) If h E BV([a, b] ) , the map x r-+ varh ( [a , x] ) is monotonely increasing. b) If h E BV, then lim varh ( [a, x] ) �f Th(x) exists, at-oo Th E MON, and lim Th(X) < 00 .

xtoo The next sequence of results leads to Lebesgue's Theorem

{I E BV} ::::} {If exists a.e. } .

The approach is due to F. Riesz . An alternative proof can be based on Vitali 's Covering Theorem .

4.6.24 LEMMA. IF f E BV([a, b] ) , FOR SOME p AND q IN MON([a, b] ) , f = (p - q) . PROOF. If a :::; c < d :::; b, then (v . 4.6.20)

.

varf ( [a , d] ) - f(d) - [varf ( [a , c] ) - f(c)] = varf ( [a , dj) - varf ( [a, c] ) - [J(c) - f(d)]

and since 0 :::; I f (c) - f(d) 1 :::; varf ( [a, dj) - varf ( [a, c] ) , both varf �f p and varf - f �f q are in MON([a, b] ) and f = varf - (varf - f) = p - q. D


4.6.25 Exercise. If I E BV, then; a) I I I is bounded: 1 1 1 1 100 :s; K < 00; b) I is the difference of two bounded functions in MON.)

The name Running Water Lemma-suggested by Figure 4.6. 1-is often given to

4.6.26 LEMMA. (F. Riesz) IF F E ]R.(O, l) , THE SET D, CONSISTING OF x FOR WHICH THERE IS SOME x' IN (x, 1) WHERE lim F(y) < F (x' ) , IS OPEN.

y=x

IF D i- (/) AND D = U (an , bn) (cf. 4.6.9) AND x E (an , bn ) , l<::n<M<::00 THEN F(x) :S; lim F(y) .

y=bn

- clef PROOF. If x' > x E D and F (x') > lim F(y) = Lx , then for some N(x) , y=x

x' tJ- N(x) , and for some N1 (x) , sup F(y) < F (x') . YENJ (x)

If y E N(x) n N1 (x) �f N2 (x) , then y < x' , and

lim F(z) :S; sup F(y) < F (x') , z=y yEN2 (x)

i .e . , N2(x) c D: D is open. If x E (an , bn ) and F(x) > Lbn , then Lx 2: F(x) > Lbn ' A contradic

tion emerges as follows . Since bn tJ- D,

(4.6.27)

By definition, for some x' greater than x, F (x' ) > Lx : F (x') > Lbn • But (4.6.27) implies x' E (x, bnl . If c �f sup { x' : x < x' :s; bn , F (x' ) > Lx } , then an < x < c :s; bn . If c < bn , then for some c' , c ' > c , and

As in the previous argument, c' E (c, bnl . However, by definition of sup, c' :s; c, a contradiction. Hence c = bn and Lbn 2: F (bn) 2: Lx > Lbn , a final contradiction. Thus F( x) :s; Lbn ' D

x-axis Figure 4.6.1 .

Section 4.6. Differentiation

[ 4.6.28 Remark. Riesz's result is applied below to the proof that monotone functions are differentiable a.e. Its analog for sequences leads to a perspicuous proof of G. D. Birkhoff's Pointwise Ergodic Theorem, [Ge3] . ]

173

lR • clef f(y) - f (x) If f E ffi. and x i- y, the ratIO gy (x) = is central to the y - x question of the existence of f'.

4.6.29 THEOREM. IF f E MaN, THEN f' EXISTS OFF A NULL SET.

PROOF. If

D1 (x) �f limgy (x), Dl (X) �f limgy (x), ytx ytx DT(x) �f limgy (x) , DT(x) �f limgy (x), ytx ytx

then 0 :::; Dl (X) :::; D1 (x) :::; 00 and 0 :::; DT (X) :::; DT(x) :::; 00.

If a) DT(x) < 00 a.e. and b) DT(x) :::; Dl (X) a.e., then a) and b) apply to x H - f( -x) , which is also in MON. Hence 0 :::; DT (x) = Dl (X) < 00 a.e. and the argument is finished once a) and b) are proved.

a) . If Eoo �f { x : DT(x) = oo }, then for each x in E, any positive A, and some y such that y > x, gy (x) > A. Hence, for the map

F : x H f (x) - Ax

and its associated set D in 4.6.26, there is a sequence { (an , bn ) } l:s;n<M:S;oo depending on A and such that D = U (an , bn) . Furthermore, l <n<M<oo E c D and - -

F ( an) :::; lim F ( x ) , A (bn - an ) :::; lim f ( x) - f ( an ) , x=bn x=bn A L (bn - an ) :::; L lim f(x) - f (an ) :::; f(l) - f(O) .

l :S;n<M:S;oo l :S;n<M:S;oo x=bn

S· E ' t ' d ' th t D ' ' (D) < f(l) - f(O) d ' Ince IS con alne In e open se , sInce /\ _

A , an SInce A may be arbitrarily large, >.(E) = O.

b) . If

E< �f { x : Dl (X) < DT(x) } , Q+ '3 r < R E Q+ , ET R = { x : Dl (x) < r < R < DT (x) } ,


then E< C U ErR. The inequality Dl (X) < r and the function {r,R}CQ

F1 : x H f( -x) + rx

lead to a set D1 �f U (an , bn ) and the inequality 1 :5n<M:5oo

( 4.6.30)

Inside each (an , bn) there is defined the function F n : x H f (x) - Rx and the possibly empty open set Dn �f U (ank , bnk ) . There is the

1 <k<K<oo corresponding inequality - -

The combination of (4.6.30) and (4.6.31) yields

L (bnk - ank ) ::::: i L (bn - an ) . k,n n

(4.6.31 )

(4 .6.32)

Induction yields a sequence {SN} NEN of sequences of open intervals. Each S N is a refinement of S N - 1 . Consequently

Since ErR C U (0 , ;3), ).. (ErR) = 0: ).. (Ed = o. D (a,(3) ES2N

The Fundamental Theorem of Calculus asserts the validity of

f(b) - f(a) = lb f' (t) dt

for suitably restricted f. In the context of Lebesgue integration there is

4.6.33 THEOREM. IF f E MON([O, l] ) , THEN: a) f' E L1 ( [O, l ] , )" ) ; b)

fs : [0, 1] '3 x H f (x) - f(O) - fox f'(t) dt IS IN BV([O, 1] ) ; c) f.� � 0; d) fs = 0 IFF f E AC ([O, 1] ) ; e)

f(x) - f(O) 2': fox f'(t) dt AND EQUALITY OBTAINS IFF f E AC( [O, 1] ) .


1 PROOF. a) If 0 < c < 1 , then for large n and x in [0, c] , x + - E [0, 1] and if n f' (x) exists, then f'(x) = nl�� n [f (x + �) - f(X)] . Hence f' E SA , and

2.1 .14a) implies lc f'(x) dx �

nl��lc

n [f (x + �) - f(X)] dx. More-over

lCn [f (x + �) - f(X)] dx = n [lc+ �

f(t) dt - l� f(t) dt]

� n [lc+ � f(c) dt - 1�

f(O) dt] � f(c) - f(O) � f(l) - f(O) :

f' E L1 ( [0, 1] , >.) . b) Both 9 : [0, 1] '3 x H f(x) - f(O) and h : [0, 1] '3 x H lx f'(t) dt are

in MON([O, 1] ) . c ) According to 4.6.18, g' � f' � h' . d) If fs = 0, then f(x) = f(O) + lx f'(t) dt, whence f E AC( [O, 1 ] ) . If

f E AC([O, 1 ] ) , then 4.6. 15 and 4.6.18 imply fs = o. e) The argument based in a) on Fatou's Lemma may be repeated to

imply f(x) - f(O) 2': lx f'(t) dt. Furthermore fs = 0 iff f E AC ( [O, 1] ) .

4.6.34 Example. a ) If

f(x) �f {�n(x) if O < x � 1 if x = 0

D

then f'(x) � .! and f' tJ- L1 ( [O, 1] , >.) . b) If f = cPo, then f' � 0, whence if x

x E [0, 1] , lx f'(t) dt = 0 and f(x) = fs (x) .

4.6.35 Exercise. If -00 < a � b < 00, assertions like those in 4.6.33 are valid for functions in BV( [a, b] ) . 4.6.36 Example. If f(x) == x , then f E MON n AC and f ' == 1 . Hence f' tJ- L1 (JR., >') , and the formula f(x) = fs (x) + [Xoo f' (X) dx is invalid.

4.6.37 Exercise. If f' exists everywhere on [a, b] and f' (a) < C < f' (b) , then for some � in (a, b) , f'(�) = C.

A derivative enjoys the intermediate value property.


[Hint: If g(x) �f f(x) - ex, then g' (a) < 0 and g' (b) > O. Hence, for some � in (a, b) , g(�) = min g(x) .] as;

xS;b

4.6.38 THEOREM. IF: a) f E ]R.[O, 1 l ; b) I' EXISTS THROUGHOUT [0, 1] ; AND c) I' E L1 ( [0, 1] , ),) , FOR EACH x IN [0, 1 ] ,

f(x) - f(O) = 1x f' (t) dt.

[ 4.6.39 Note. The result is in sharp contrast with 4.6.34e) . Thus the assumption b) provides the crucial ingredient for the validity of the conclusion.]

PROOF. The line of argument is to show that f(l) - f(O) � 11 I' (t) dt.

That conclusion is applicable to -f and leads to f(l ) - f(O) = 1x t(t) dt. The same kind of reasoning can be used when 1 is replaced by any x in [0, 1] .

Although, as a derivative, I' enjoys the intermediate value property, that alone does not suffice for the present purposes. In [GeO] there is an example of a null function that assumes every real value in every subinterval.

The technique is rather to exploit the lower semicontinuity of an auxiliary lsc function 9 that approximates I' from above.

The DLS construction implies that if E > 0, there is in lsc( [O, 1] ) a 9 1 1 1 such that 9 2': I' and ° g dt < 1 I' dt + 2E. Hence 9 + E �f P E lsc([O, 1] ) ,

p > 1' , and 11 p dt < 11 I' dt + E .

If, for every positive E , 11 p(t) dt - f(l ) + f(O) + E 2': 0, then

f(l ) - f(O) :::; 11 p(t) dt + E < 11 f'(t) dt + 2E,

whence f(l ) - f(O) :::; 11 I' (t) dt. The desired conclusion is reached via

the following argument showing that if

G, (x) �f 1x p(t) dt - f(x) + f(O) + EX,

then for every positive E, G, (l) 2': O.

Section 4.7. Derivatives 177

Since G, (O) = 0, if G,( l ) < 0, for some largest x in [0, I ) , G, (x) = 0, i.e.,

{x < x' :::; I} ::::} {G, (x') < O} . (4.6.40) The hypothesis b) implies that f' (x) exists: for some positive h, x + h < 1, and if x :::; y :::; x + h, then f(y) - f(x) < f'(X) (Y - x) + E. Since p is lsc, for some positive k, x + k < 1 and if x :::; t :::; x + k, then p(t) > J' (x) . Hence, if J �f min{h, k} « I ) , then

lx+8 G, (x + J) = G, (x + J) - G, (x) = x p(t) dt - f(x + J) + f(x) + EJ

> J'(x)J - [J' (x) + E] J + EJ = 0,

in denial of (4.6.40) . [ 4.6.41 Remark. Many more theorems about derivatives can be found in the literature, e.g. ,

• (Denjoy-Young-Saks) If f E ffi.lR , for some set A, )'(A) = 0, and if x tJ- A, then

-lim- f (x + h) - f (x) = 00 = _ lim _f (-,---x_+_h )'----_f-'...( x--'-) h"tO h h"tO h '

or f' (X) exists,

v. [SzN] ; • In the metric space C ( [O , 1 ] , ffi.) , the set of nowhere differen

tiable functions is of the second category, cf. 3.62 in [Ge3] .]

4.6.42 Exercise. If f E BV and for some g, f = g' , then f E C(ffi., C) .

4.7. Derivatives

D

The developments in Section 4.6 lead to a rigorous proof of the change of variables formula of multidimensional calculus . n

F clef ) d clef ( ) . mm ( clef '" or x = (XI , . . . , Xn an y = YI , . . . , Yn In m. , x, y) = � XkYk k=1 n

and I lx l l � �f L x� . k=1

4.7. 1 DEFINITION. A FUNCTION f : ffi.n '3 x H Y �f f(x) E ffi.m IS differentiable at a IFF FOR SOME Ta IN [ffi.n , ffi.m] ,

lim I l f(a + h) - f(a) - Ta(h) 1 1 2 h -+ O I Ih l 1 2

= 0; h>' O


T. d " I T.a <L_ef I' (a) . a IS THE envatwe OF AT a:

4.7.2 Exercise. If I' (a) exists, then I is continuous at a.

4.7.3 Exercise. If m = n = 1 and I E BV is of bounded variation, then I'(a) is the Radon-Nikodym derivative of I with respect to A. 4.7.4 Exercise. If I E [lR.n ] , then I' == I, i.e . , the linear transformations I' and I are the same. 4.7.5 Exercise. a) If 0' i- v' E (!R.n ) ' then

An ({ v : (v, v' ) = O } ) �f An (H) = O. b) If M is a vector subspace of !R.n and dim (M) = k < n, then An (M) = O.

[Hint: a) For any C . . . , if E �f H n C . . . , Fubini's Theorem implies le I dAn = O . b) Each M is the intersection of finitely many hyperplanes like H . ]

4.7.6 Exercise. a) If I E [!R.n , !R.m ] and E E SAn ' then I(E ) E SA= ' b) If m = n, the map f..l : SAn '3 E f--t A[/(E)] is a possibly trivial translationinvariant measure. c) For some nonnegative constant p(f) ,

f..l(E ) == p(f)A (E) and Df..l(x) == p(f) . [Hint: For an open cube C . , I (C, .) is open or a null set .] The result in 4.7.6 motivates

4.7.7 THEOREM. IF U IS AN OPEN SUBSET OF !R.n , x E U, I E C (U, !R.n ) , AND I' (x) EXISTS, THEN I (B(x, rn E S,(3 (!R.n ) AND

lim A [I (B (x, r ) ° ) ] = p [I' ( x )] . T--+O A [B(x, r)o ] (4.7.8)

PROOF. Since I (B(x, r) O ) = U I {B [x, (1 - Tn) r] } and each summand nEN

in the right member above is compact , I (B(x, rt ) is a-compact , and hence in S,(3 (!R.n ) .

Since A is translation-invariant , the conclusion when x = 0 = 1(0) implies the general case. The line of the argument depends on whether the linear map 1' (0) �f T is : a) nonsingular ; or b) singular .

For a) , if 9 �f T- 1 0 I, then g' (0) = T- 1 I' (0) = id , and the desired equality is

lim A {g [B(x, r)O] } = 1 T--+O A [B(x, r)o ] , (4 . 7.9)


v. a') after the PROOF of 4.7. 10. For b) the range im (T) is a subspace of dimension less than n. Thus

(4.7.5) im (T) is a null set (A) , whence p(T) = ° and the desired result is that the left member of (4. 7.9) is zero, v. b ' ) after the proof of a' ) .

When x = ° = 1(0) and T = id the numerator of the basic difference quotient is 1 1 / (x) - x1 1 2 ' which motivates

4.7. 10 LEMMA. IF g E e (B(O, 1 ) , ]R.n ) , E E (0, 1 ) , AND

{ l l x l 1 2 = I} '* { l l g(x) - xl 1 2 < E} , THEN 9 (B (0, 1 n ::) B (0, 1 - E t . PROOF. If the conclusion is false, for some y,

I I Y l 1 2 < 1 - E and y tJ- g (B(O, ln ,

(4 .7. 1 1 )

a contradiction. Since I I Ix l 1 2 - I lg(x) 1 1 2 1 � I l g(x) - x1 1 2 , (4 .7 . 1 1 ) implies that if I Ix l 1 2 = 1 , then I l g(x) 1 1 2 > 1 - E > I ly 1 1 2 , i .e. , not only is y not in 9 (B(O, In but y is not in g[B(O, 1) ] . Thus G : x

H I I y - g

/X)I I IS a y - 9 x) 2

well-defined continuous map of B(O, 1 ) into itself. Brouwer's Fixed Point Theorem ( 1 .4.27 and 1.4.36) implies that

for some xo , G (xo) = Xo. If I I xo l 1 2 = 1 , the equation G (xo) = Xo and Schwarz's inequality [(3.2. 11 ) ] imply

I l y - 9 (xo) 1 1 2 = (xo , Y - 9 (xo ) ) = (xo, y) + (xo , x - 9 (xo) ) - 1 0, for a positive J,

As a consequence, 1 1 /(x) 1 1 2 < (1 + E) l l x I 1 2 ,

i.e . , 1 [B(O, rt] C B[O, (1 + E)r] o . By virtue of 4.7.10, if E E (0, 1 ) , then B(O, ( 1 - E)rt C 1 (B(O, rn.

A {J [B(x, r )O ] } Hence, if E E (0, 1 ) , then (1 - E)n � A [B(x, r)O ] � (1 + Et and (4.7.9) obtains.

b' ) (T is singular) If E > ° and U, �f { x : inf I l x - Y l 1 2 < E } ' yElm (T)

then U, is the open set consisting of all x within a distance of E from some point in im (T) . If K is compact, K n U, is covered by finitely many


open balls of radius E. Hence, for a constant C K , A (K n U, ) :::; C KEn . Furthermore, if Kl C K2, then CK, :::; CK2 . For a positive ro , if r :::; ro and I Ix l 1 2 < r, there is an Ar such that I l f (x) - J'(O)xI 1 2 :::; Ar l lx l 1 2 , and lim Ar = O. The facts just stated imply that when Kr is the compact set rtO f[B(O r)] A {J [B(O , r)O ] } < ArCKrrn = ArCKr . D , , A [B(O, r)o ] - rn

[ 4.7. 12 Note. In (4.7.9) each B(x, rt may be replaced by an open cube Cr(x) with edges parallel to the coordinate axes, edgelength 2r, and containing x. Corresponding to (4.7.9) is

1· A {g [Cr (x)] } = [f' ( ) ] r� A [Cr (x)] P x .

There is a constant Kn such that for all x in ffi.n , the norm

satisfies I lx l l ' :::; I I x l 1 2 :::; Kn l lx l l ' , i.e. , II I I ' and II 1 1 2 are equivalent . The open ball centered at x and of radius r for the norm I I I I ' is

B' (x, rt �f { y : I lx - y l l ' < r } ,

i .e . , the open cube with edges parallel to the coordinate axes, edge-length 2r, and centered at x. Corresponding to 4.7. 10 is the statement:

If 9 E C (B' (O, 1t , ffi.n ) , E E (0, 1 ) , and

{ l lx l l ' = I} '* { l l g(x) - x i i ' < E} , then 9 (B' (0, 1 n ::) B' (0, 1 - E t .]

4.7. 13 Exercise. The assertions in 4.7. 12 are valid. Together with 4.7.6, 4.7.14 below forms the basis for the change of

variables formula of multidimensional calculus (4.7.23 below) .

4.7.14 LEMMA . IF N IS A NULL SET (A) IN ffi.n , f E (ffi.n )N , AND FOR EACH Y IN N, inf sup I l f (

I�) -ffr) 1 1 2 < 00, THEN f(N) IS A NULL

N(y) N(y)3x#-y x - Y 2 SET (A ) .


PROOF. Since N is the union of the count ably many sets

Ekp �f { X : x E N, I l f (x) - f(y) 1 1 2 � k l lx - y I 1 2 , Y E B (x, �) n N } , k, p E N,

if each f (Ekp) is shown to be a null set (), ) , the result follows. For the rest of the argument, the subscript kp is dropped.

If E, 1] > 0, E is contained in an open set U such that ), (U) < 1]. As in the PROOF of 4.6.8, U is the union of pairwise disjoint half-open cubes Q . . . and L), (Q . . . ) < 1]. Since each half-open cube Q . . . is the union of

. . . all the smaller half-open cubes i t contains, E (= Ekp) can be covered by pairwise disjoint half-open cubes of diameter less than � . If x . . . E E n Q . . . , p then Q . . . C B [x . . . , diam (Q . . . )t , whence for some constant K, there obtains for the sum of the measures of all the open balls B [x . . . , diam (Q . . . W corresponding to all the Q . . . in all the Q . . .

L), {B [x . . . , diam (Q . . . )t } = KL), (Q . . . ) < K1].

Hence, if 1] = �, then L), {B [x . . . , diam (Q . . . )t } < E.

According to the definition of E (= Ekp) ,

f(E) C UB [f (x . . . ) , k · diam (Q . )] ,

whence

L), {(B [f (x . . . ) , k · diam (Q . . . ) ] } � kn L), {B [x . . . , diam (Q . . . )t }

D

4.7. 15 COROLLARY. IF V IS AN OPEN SUBSET OF ffi.n AND f' EXISTS AT EACH POINT OF V, THEN: a) f MAPS NULL SETS (), ) CONTAINED IN V INTO NULL SETS (),) CONTAINED IN f(V) :

{ {,>, (N) = o} 1\ {N C V}} '* {,>, [f(V)] = o} ; b) f MAPS LEBESGUE MEASURABLE SETS INTO LEBESGUE MEASURABLE SETS: {E E S),} '* {J(E) E S),} . PROOF. a) 4.7. 14 applies.


b) Since ), is regular and JR.n is a countable union of compact subsets (JR.n is a-compact) , if E E S)" then for some a-compact set 5 and a null set N, E = 5'0N. Hence f(E) = f(5) U f(N) . Moreover, f(5) is a-compact ; owing to a) , ), [f(N ) ] = O. D

4.7.16 Exercise. There is no conflict between 4.7.15 and the following facts: a) the Cantor function cPo is differentiable a.e. (),) ; b) the Cantor set Co is a null set (),) ; c) ), [cPo (Co)] = l .

4.7.17 Exercise. a) If f E (JR.n )lR= f (x) �f (/1 (x), . . . , fn (x) ) and J' ex-. h 8 fi (x) . 1 < . < 1 < . < d h ' . Ists, t en -8-- eXIsts, _ l _ n, _ J _ m, an t e matnx representmg Xj the linear transformation J' with respect to the standard basis el , . . . , en is

J' (x) = (8fi (X) ) �f J(I) , 8xj ISiSn, ISjSm the Jacobian matrix of f. b) The entries in the Jacobian matrix J(I) are Lebesgue measurable functions. c) If f E [lRm , JR.n ] (c (JR.n )lR= ) and

n f (ei ) �f L aijej , 1 � i � m, j"" l

8fi (X) . . then aij = -8-- ' 1 � l � n, 1 � J � m. X · J

4.7.18 THEOREM. IF: a) f E (JR.n )B(O, 1 ) o ; b) J' EXISTS THROUGHOUT B(O, It ; c) f IS INJECTIVE AND sup { l l f(x) 1 1 2 : x E B(O, It } < 00;

d) g : JR."' f--t [0, (0) IS LEBESGUE MEASURABLE; e) p IS THE FUNCTION IN 4.7.6 , THEN BOTH P (I' ) 1 B(O, I ) O AND g o f . p (I' ) 1 B(O, I) O ARE LEBESGUE

MEASURABLE AND J 9 d)' = r (g 0 f) . p (I' ) d)'. f(B(O, I) O ) iB(O, 1 ) 0 PROOF. The measurability of p (l' ) I B(o, 1 ) 0 is a consequence of the results in 4.7.17.

If E E S)" 4.7.15 implies: a) p,(E) �f ),[f(E)] exists; b) (JR.n , S)" p,) is a complex measure space (direct calculation) ; c) p, « ),. Owing to 4.6. 12 , if E E S), n B(O, It , then p,(E) = Ie Dp, d)'. Hence, for x in B(O, It and small r,

p, [B(x, rt] ), [B(x, r)o]

), [f (B(O, r)O) ] ), (B(x, r)O )

Owing t o 4.7.6 and (4.7. 19) , Dp,(x) � p [J' (x) ] (v. 4.7. 12) .

(4.7.19)


If A E S,B and E �f f- 1 (A ) , then XE = XA 0 f: E E S,B (C SA ) . Thus

J X d)" = r X 0 f . p (I' ) d)". f(B(0, 1) 0 )

A lB(0, 1 ) 0 A ( 4 .7.20)

If N is a null set ().. ) , for some A in S,B, A ::) N and )"(A) = o. Thus (4.7.20) is valid with A replaced by any Lebesgue measurable subset E of B(O, It . The standard approximative methods extend the validity of (4.7. 20) to the case where X is replaced by an arbitrary nonnegative

A Lebesgue measurable g. D

[ 4.7.21 Note. The PROOF of 4.7.18 shows that g o f . p (I' ) is measurable; g o f need not be measurable (cf. 4.5.8) . ]

4.7.22 Exercise. If: a) V is open in ffi.n ; b) X is a Lebesgue measurable subset of V; c) f : V '3 x f--t ffi.n is continuous, f is injective throughout X, and f' exists throughout X; d ) )"(V \ X) = 0 ; and e ) 9 : ffi.n f--t [0, (0) is Lebesgue measurable, then

J 9 d)" = r 9 0 f . p (I' ) d)". f(V) lv

[Hint: The open set V is the countable union of open balls; 4.7.18 applies.]

4.7.23 THEOREM. IF f E [ffi.n] AND f' EXISTS, THEN

p [!, (x)] = Idet {J [f(x) ] } I .

PROOF. If T �f !' (x), e is a constant, and

then Idet {J [f (x) ] } I = l ei .

if k = i otherwise '

On the other hand, if the edge-lengths of a cube C . . . are S1 , • • • , Sn and the corresponding edge-lengths of T ( C . . . ) are s� , . . . , s� , then sj = s j if j i- i, and s ; = l e i · Si , whence p(T) = l e i : Idet {J [f(x) ] } 1 = p(T) .

If if k sl {i , j} if k = i if k = j


T(ABCD) = ACED

A(ABCD) = A.(ACED), - " E

, , , ,

D t-C-----+----.,.,,' C

A B Figure 4.7. 1 .

then l[f(x)] i s the result of interchanging rows i and j of the identity matrix: Idet {l[f(x) ] } 1 = 1. Because the measure of any cube C . . . of edgelength £ is £n, regardless of the labeling of the coordinate axes, p(T) = 1 : Idet {J [f (x) ] } I = p(T) .

If T (ek) = { ei + ej if k = i

ek otherwise '

then Idet {J [f (x)] } I = 1 . If n = 2, by direct calculation (integration or elementary geometry) , ),2 (C . . . ) = ),2 [T (C . . . ) ] . The situation is illustrated in Figure 4.7.l .

If n > 2, via Fubini' s Theorem and induction, ),n (C . . . ) = ),n [T (C . . . ) ] : Idet {J [f (x) ] } I = 1 = p(T) .

Since every linear transformation is the composition of a finite number of transformations, each like one of the three just described, the product rule for determinants applies [Ge2] . D

4.7.24 Exercise. a) If a2 + b2 > 0, the map

is bijective and !, exists. b) What is p (I' ) ? When (X, S, p,) and (Y, T, �) are measure spaces an F in yX i s defined

to be measurable iff {E E T} ::::} {J- l (E) E S} [Halm] . In the current 2 context, since [ffi.n ] may be regarded as ffi.n , !' as a function of x may be ]Rn

regarded as an element of (ffi.n2 ) 4.7.25 Exercise. Both f' and p (I' ) are Lebesque measurable.

Section 4.8 . Curves 185

4.8. Curves

4.8 .1 DEFINITION. A curve IN A TOPOLOGICAL SPACE X IS AN ELEMENT "/ OF C([O, 1] , X) . THE SET im h) �f "/* �f { "/ (t) : t E [0 , 1] } IS THE image of"/. WHEN X IS A METRIC SPACE (X, d) , THE (POSSIBLY INFINITE) length fih) OF "/ IS

WHEN fih) IS FINITE , "/ IS rectifiable . WHEN ,,/(0) = ,,/(1 ) , "/ IS closed. WHEN "/ �f hI , . . . , "/n ) E C ( [O, 1] , ffi.n ) , f E C h* , c) , AND "/ IS RECTIFIABLE, THE Riemann-Stieltjes integral of f with respect to "/ IS

4.8.2 Example. For the curves

"/1 : [0, 1] 3 t f--t e27rit E C, "/2 : [0, 1] 3 t f--t e37rit , "/3 : [0, 1] 3 t f--t e47rit , "/� = "/� = ,,/; , while fi hd = 271", fi (2) = 371", fi (3) = 471". Furthermore, "/2 is not a closed curve.

4.8.3 Exercise. For the Cantor function <Po , if ,,/o (t) = t + i<po (t) , then fih) = 2. For a in [0, 1 ) , the Cantor function <Pc » and the curve

what is fi ho: )?

4.8.4 Exercise. If

then fih) = 00 .

"/0: : [0, 1] 3 t f--t t + i<Pa (t) ,

if ° < t ::; 1 if t = °

4.8.5 Exercise. For a rectifiable curve "/ : [0 , 1] 3 t f--t ,,/(t) E ffi.n and a Riemann partition P �f {tdl:S:k:S:n of [0, 1] : a) the formulre

"/P [Otk + (1 - O)tk+d = 0,,/ (tk ) + ( 1 - 0)"( (tk+d ° ::; 0 ::; 1 and 1 ::; k ::; n - 1 ,


define a rectifiable curve, "/ p and £ ("! p) ::; £ (,,/ ) ; b) if E > 0, then for some P, £ ("!p» £("!) - E; c) {£ (,,/p » £("!) - E} ::::} { lhp - "/ l loo < E} . d) if n = 2,

f E C ( [0, 1] , C) , and E > 0, for some P, I II f d,,/p - 11 f d"/ I < E. [ 4.8.6 Note. The curve ,,/p is piecewise differentiable ; on each interval (tk , tk+d , "/� is constant. The set b (tk ) } l <k<n of its vertices is a subset of ,,/* . If f E C( [O, 1] , C) , then

- -

More generally, the calculation of 1 f d,,/ is eased if "/ is wellbehaved or, like ,,/P, only piecewise well-behaved. For example, if {tk} l:$k:$n is a Riemann partition of [0, 1] and

then 1 f dz = 11 f 0 "/ . "/' dt.]

4.8.7 THEOREM. IF ,,/ �f a + ib, {a, b} c C ( [O, 1] , ffi.) , THEN "/ IS RECTIFIABLE IFF {a , b} C BV([O, 1] ) . PROOF. The triangle inequality (v. 3.2.12) implies that

max { Ia (tk+d - a (tk ) 1 , I b (tk+d - b (tk ) l } ::; b (tk+d - "/ (tk ) 1 ::; l a (tk+d - a (tk ) 1 + I b (tk+d - b (tk) l ·

Thus "/ is rectifiable iff {a, b} c BV ( [0 , 1] ) . [ 4.8.8 Note. If "/ is rectifiable, then £("!) is not only

{ n- l } sup L h (tk+d - "/ (tk ) 1 : 0 = to < tl < ' " < tn = 1 , n E N ,

k=O i.e. , the supremum of the lengths of the associated polygons with

vertices on "/* but, by abuse of notation, £("!) = 11 1 · I d"/ I .]

D

Section 4.9. Appendix: Haar Measure 187

4.9. Appendix: Haar Measure

A topological group G is a Hausdorff space and a group such that

G X G '3 (x, y) H xy- l E G

is continuous. For a locally compact group G, there is a measure - Haar measure-p, defined on aR[K(G)] �f S(G) . For p, there obtains:

{{ E E S(G) } !\ {x E G}} '* {{ xE E G} !\ {p,(xE) = p,(E) } } , i.e. , p, is translation-invariant .

Complete proofs of the existence of Haar measure p, and derivations of its most important properties are given in [Halm, Loo, Nai, We2] . Below is an outline of the fundamental idea behind the existence proof.

a) The motivation lies in the next alternative definition of the Riemann integral. If f E Coo (ffi., ffi.+ ) and Nf E N there is a non empty family C of sets {cn} -Nt ::;n::;Nt of nonnegative constants such that

(2Nt ) 2 _ 1 f(x) � L

k=O Nt

Furthermore, 1 f(x) dx = i�f L cn· The integral stems from the lR n=-Nt

majorization of f by linear combinations of translates of x . b) Haar's idea was to imitate the procedure described in a) , namely to

majorize one function by linear combinations of translates of another and thereby approximate some kind of DLS functional. For a locally compact group G and a pair {f, g} in Coo (G, ffi.+ ) there is a (possibly empty) family C of sets {cn} -N <n<N of nonnegative t - - t constants such that for some subset S �f {Yn} -Nt ::;n::;Nt of G

Nt f(x) � L cng (Ynx) .

n=-Nt Nt

If g =t=- 0, then C i- (/) and (f : g) �f i�f L Cn E ffi.+ . The functional n=-Nt

(f : g) enjoys the following properties. (All functions below are in


Coo (G, ffi.+ ) ; neither 9 nor h is identically zero; f[y] (x) �f f (yx) . )

(f[y] : g) = (f : g) , (f + k : g) ::=; (f : g) + (k : g) , {t 2: o} '* { (tf : g) = t(f : g) } , {J ::=; k} '* {(f : g) ::=; (k : g) } ,

(f : h) ::=; (f : g) . (g : h) ,

(f ' ) > I l f l loo . 9 - I l g l loo '

For a fixed nonzero fo in Coo (G, ffi.+ ) , if Ig (f) �f ? :. g\ , then fo . 9

1 (fo : g) ::=; lg (f) ::=; (f : fo ) .

(4 .9 .1 )

• The choice of fo is arbitrary. Once fixed, fo sets the scale of the measure.

• It is to be expected that if 9 ::=; h, then Ig approximates the desired functional more satisfactorily than h . The set Coo (G, ffi.+ ) is a poset with respect to )- , as defined by: {g )- h} {} {g ::=; h} (smaller functions succeed larger functions) .

• If f E Coo (G, ffi.+ ) , since [ 1 ) ' (f : fo)] is compact, for the (fo : g

net n : Coo (G, ffi.+ ) 3 g r---t 1g (f) E [ (fo1: g) , (f : fol there is a

convergent cofinal net . Its limit is defined to be 1(f) . Owing to (4.9 . 1 ) , I is a translation-invariant linear functional:

I (f[y] ) = 1(f) .

c) Arguments using partitions of unity lead to the conclusion that I can be extended , say to J, a DLS functional having as domain Coo ( G, ffi.) . The machinery of Chapter 2 applies and produces the (left ) translation-invariant Haar measure p, defined on S �f aR[K(G) ] : if E E S and x E G, then p,(xE) = p,(E) . The following are the basic facts about Haar measure:

• if V is a nonempty open set in S, then p,(V) > 0; • if K is compact , p,(K) < 00;

• if � is a (left translation-invariant) Haar measure, then for some positive constant c, � = cp,; c depends upon the choice of fo in the construction above.

Section 4 .9. Appendix: Haar Measure 189

For the locally compact groups ffi. resp. ffi.n resp. 'lI' resp. Z, A resp. An resp. T resp. V are (the essentially unique) translation-invariant measures .

The convolution of two functions f and g in Ll (G, p,) is defined by

With respect to pointwise addition and multiplication defined as convolution, Ll (G, p,) is a Banach algebra A(G), the group algebra of G. The group algebra: a) is commutative iff G is abelian; b) has an identity iff G is discrete.

If G is abelian, for the measure � : 5 '3 E H P, (E- l ) , �(xE) = p, ((XE) - l ) = P, (x- l E- l ) = p, (E- l ) = � (E) .

If E = E-\ then �(E) = p,(E) . Since Haar measure is unique up to a multiplicative constant, � = p,.

The following observations are valid when G is abelian. 1) If

clef ' M E Sp [A(G)] ' f E A(G), a E G, f[a] (x) = f (ax) , and f(M) i- 0,

then {[a] (M)

aM : G '3 a H -'-;;:,-'------f(M) (4.9.2)

is independent of the choice of f and aM is a continuous open homomorphism of G into 'lI'. Furthermore, G denoting the character group of G, i.e. , the set of all continuous open homomorphisms of G into 'lI',

g : Sp (A) '3 M H aM E G

is a bijection. As in the case of Banach spaces and their duals, if "( E G the notation (x, "() is used for ,,((x) .

2) Since G and Sp [A( G)] are identified, the latter inherits a topology via a (A', A) for A' . Since the unit ball of A' is a (A', A)-compact, G is a loc�lly compact abelian group. If x E G, the,n hx : G '3 "( H (x, ,,() E 'lI' is in G. Moreover (Pontrjagin) G '3 x H hx E G is an isomorphism (in the category of topological groups and continuous open homomorphisms) .

3 ) If f E A(G), the Fourier transform j, also called the GelfandFourier transform, may be viewed as a function on G. For j there ob-tains the formula: ' : A(G) '3 f H i f (x) (x, "() dp,(x) �f jb) . Further-

more, j E Co ( G, C) , and since the quotient map is a norm-decreasing

homomorphism, 1 11 1 100 � I lf l l l '


4) If {f, g, h} c Coo (G, q , Holder's inequality and the translationinvariance of p, imply

(4.9.3)

On the other hand, f E £P(G, p,) and h E £P' (G, p,) . Hence, if I l h l l p' � 1, Fubini's Theorem implies

Ii [i f (y- 1 X) 9(Y) dP,(Y)] h(x) dp,(x) I = I i (J[y] , h) g(y) dp,(y) I � I I f[y] l ip . I l g l l l '

Thus 3.3.6 implies (4.9.4 )

5) Equation (4.9.3) resp. (4.9.4) may be extended and interpreted as follows:

If 9 E Coo (G, JR.) , the map

Tg : Coo (G, JR.) '3 f H f * 9 E Coo (G, JR.)

is defined on the II l ip-dense subset Coo ( G, JR.) of

£P(G, p,) , 1 � p < 00.

According as Coo (G, JR.) is viewed as a subspace of the Banach space Co(G, JR.) resp. LP(G, p,) , 1 � p < 00, Tg may be normed:

resp.

l iT II �f I ITg (f) l loo - II I I �f M 9 - sup - 9 1 - ' 0 00 I lf ll p,to II f l ip p

l iT I I �f I ITg (f) l lp - II I I �f M 9 - ,>up - 9 p' - , , . p I l f l l p,tO II f l i p p p

4.9.5 Exercise. Under Tg , I I I Ip-Cauchy- sequences map into I I 1 100-Cauchy sequences.

[ 4.9.6 Note. By continuity, Tg may be defined uniquely on all Co(G, q resp. LP(G, p,) , 1 � p < 00. Furthermore, Tg is definable for 9 in L 1 (G, p,) resp. LP' (G, p,) . In all these elaborations, the numbers M.1o resp. M.1 .1 remain unchanged.]

P P P

The following extension of (4.9.4) is due to W. H. Young.

Section 4.9. Appendix: Haar Measure

. { } 1 1 clef 1 4.9.7 LEMMA. IF 1 :::; mm p, q , - + - - 1 = - > 0, AND P q r

(I, g) E U(G, /-l) X Lq (G, /-l) ,

THEN I l f * g i l T :::; I l f l lp . I lg l l q · 1 1

[ 4.9.8 Remark. If q = p' , then - + - = 1 and, by abuse of noP q tation, r = 00. Then the conclusion of 4.9.7 reduces to (4 .9.4) . ]

191

P Th d· · 1 1 1 clef 1 . 1 11 . 1 h ROOF. e con Itlon - + - - = - > 0 IS ogica y eqmva ent to t e P q r

assumption that there are positive numbers 0 , (3, "I such that

1 1 1 1 1 1 1 1 1 - + - + - = I , , 0 = r. o (3 "I P 0 (3 ' q 0 "I

Thus the factorization

I f (y- l X) 1 ' lg(y) 1

(4.9 .9)

= ( I f (y- lX) I ;; Ig(Y) I � ) ' ( I f (y- lx) I P( � - ± )) . ( lg (y) l q ( i - ± ) ) (three factors) and 3.7.16 when 0 = PI , (3 = P2 , "I = P3 imply

1

I f * g(x) 1 :::; [L I f (y- lX) I P .* Ig(y) l q d/-l(y)] ;;

1 . [L I f (y- l x) I Pi3( � � ± ) d/-l(y)] i3

(4.9.10)

Of the last two factors in the right member of (4.9. 10) , the former is independent of x (because /-l is translation-invariant) and is II f l i p ; the latter is I lg l l q . The translation invariance of /-l implies also that

Direct calculations using the relations (4.9. 10) lead to the result. D [ 4.9. 11 Note. The derivation above is based on nothing but careful use of Holder's inequality and its extension. In 1 1.2 .13 the same result is seen to be one of several consequences of the M. Riesz Convexity Theorem, 11 .2.6, of fundamental importance


in the study of Fourier series and Fourier transforms , the basic ingredients of harmonic analysis on locally compact abelian groups [Loo, We2] .]

4.9.12 Exercise. a) The set C* �f (C \ {O}) is a locally compact abelian group when its topology is that inherited from the customary topology of C and multiplication (of complex numbers) is the group binary operation. b) The map f.-l : S), (C* ) 3 E H r 2

1 2 dx dy is a Haar measure for C* . JE x + y

[Hint: The discussion in 4.7.24 applies.]

4 .10. Miscellaneous Exercises

4.10.1 Exercise. (The Metric Density Lemma) If E E S)'n ' then

1. ). [E n B(x, r) ] 1m ----'-:--;-:=-:--'-..,.,--'-'-rtO ). [B(x, r) ]

exists a.e. The limit is 1 for almost every x in E and is ° for almost every x in ffi.n \ E.

[Hint: The results 4.6.11 , 4.6. 14-4.6. 17 apply when f �f XE.]

4.10.2 Exercise. If n = 1 , 4.7.18 can be proved without recourse to Brouwer's Fixed Point Theorem. 4.10.3 Exercise. If {an}nEN C ffi., {dn}nEN C (0, 00), and

00

if t < a if t 2: a '

then f �f L dnjan is in MON and Discont (I) = {an} nEN' n=l For a in (0, 00) and a Lipschitz function (v. 3.2.31)

L : ffi.2 3 {x, y} H L(x, y) E (0, 00) ,

an f in ffi.lR i s in Lip (L, O') iff I f (x) - f(y) 1 :::; L(x, y) lx - y in .

4.10.4 Exercise. a) If L is a constant and f E Lip (L, 1 ) , then f E AC. The converse is false. b) If f E Cl ( [O, 1] , ffi.) , for some constant Lipschitz function L, f E Lip (L , 1) . The converse is false. 4.10.5 Exercise. If {J, g} c AC, then {J ± g, fg} C AC. For some f in AC and some g in Coo (ffi., C) , fg tJ- AC .

Section 4. 10. Miscellaneous Exercises

4.10.6 Exercise. a) If 1 E AC ( [a, b] ) and 9 E C( [a, b] , q , then

ib 9 dl = ib 9 . f' d).. .

b) If 1 (x) �f {O �f -00 < x :::; 0 , 1 1f O < x < 00

9 E C(JR., q , and g(O) = 1 , then 1 E BV and

1 = l 9 dl > l 9 . f' d).. = O.

c) For the (continuous) Cantor function cPo :

193

4.10.7 Exercise. If 9 E C([a, b] , q and h E BV([a, b] ) , then: a) ib g dh exists; b) the following formula for integration by parts is valid:

[Hint: a) Only the situation 9 E C([a, b] , JR.) , h E MON( [a, b] ) need be addressed. b) The formula (Abel summation)

n-1 L 1 (�k ) [g (Xk+ I ) - 9 (Xk ) ] k=1

n-1 = 9 (xn) 1 (�n- I ) - 9 (xI ) 1 (6 ) - L g (Xl,; ) [I (�k ) - 1 (�k- I ) ] k=2

applies to the Riemann sums approximating ib 1 dg . ]

4.10.8 Exercise. If 1 E C( [a, b] , JR.) , 9 E BV([a, b] ) , and

F(t) = it 1 dg, t E [a, b] ,

then: a) F E BV([a, b] ) ; b)

F(t) - F(t-) = I(t) [g(t) - g(t-) ] , F(H) - F(t) = l(t) [g(H) - g(t) ] ;


c) F' � f · g' .

4.10.9 Exercise. If U E 0 (JR.n ) , for a sequence {Kn}nEN of compact subsets: a) U Kn = U; b) Kn C K�+l c) if K (JR.n ) 3 K c U, for some N,

nEN N

K c U Kn. n=l [Hint: For each n and m in N, the union of the finitely many Rk,m (v. Section 4.6) contained in B(O, nt n U is a compact subset of U. If K is a compact subset of U, and d is the Euclidean metric for JR.n , then inf { d(x, u) : x E K, u E U } > 0.]

clef ) 4.10.10 Exercise. If "Ii E AC ( [O, 1] ) , 1 � i � n, and "I = ("11 , . . . , "In , then flh) = 11 11"1' 1 1 2 dt.

4.10. 11 Exercise. If G is a topological group and H is an open subgroup of G, then H is closed (whence, if H ¥-G, G is not connected) .

[Hint: Each coset of H i s open.]

4.10.12 Exercise. The value of aM in (4 .9 .2) is independent of the choice of f. 4.10.13 Exercise. If G is a locally compact group with Haar measure p, and for each neighborhood V of the identity of G: a) Uv is a nonnegative function in A(G); b) Uv = ° off V; and c) i uv (x) dp,(x) = 1 , the net

V H Uv * f converges [in A(G)] to f and (uv )a (a ) as a function of a, converges uniformly to (a, a ) .

[ 4.10.14 Note. The Banach spaces Ll ( [O , 1 ] , >.) and Ll ( [O, 1 ) , >.) are essentially indistinguishable since >. ( {I} ) = 0. The map

is a continuous bijection between [0, 1 ) and 'lI'. The topology T �f { E : E C 'lI', \11-1 (E) E O{ [0, I )} } is that inherited by 'lI' from JR.2 and with respect to T, 'lI' is a topological group. The measure spaces ( [0, 1] ' 5,6( [0, 1 ] ) , >.) and ('lI', 5,6 ('lI') , T) are isomorphic via the bijection \II . Thus Ll ( [O , 1] , >') and Ll ('lI', T) are isomorphic in the category of Banach spaces and continuous homomorphisms.]


4.10.15 Exercise. With respect to convolution as multiplication, i .e . , with respect to the binary operation

the Banach space L1 (1I', T) is a Banach algebra A(1I') . 4.10.16 Exercise. If G is a locally compact group, f..l is Haar measure, p 2: 1 , E > 0, and I E U(G, f..l) , in N(e) there is a V such that

4.10.17 Exercise. If {FN} NEN is the set of Fejer's kernels (v . 3.7.6) and I E U([O , 1 ] , ), ) , then FN * I exists for each N and

lim I IFN * I - I l lp = O. N-too

[Hint: The result is true if I E C( [O, 1] , q .]

4.10.18 Exercise. If {J, g} c (ffi,m )lRn and both I' and g' exist at some a in ffi,n , then: a) h �f (f, g) E ffi,lRn ; b) h' exists at a; c)

[Hint:

h' (a) = (f, g) ' (a) = (f' (a) , g) + (f, g' (a) ) .

(f(a + x) , g(a + x) ) - (f(a) , g(a) )

= (f(a + x) , g(a + x) ) - (f(a + x) , g(a) )

+ (fa + x) , g(a) ) - (f(a) , g(a) ) .]

4.10.19 Exercise. If E c ffi, and )'(E) = 0, then ffi, \ E is dense in ffi.. 4.10.20 Exercise. For

1 m2 ( ) { (xi - x�) if X21 + x22 > 0 : m. '3 X1 , X2 H

O(xi + x� )

otherwise

how do the iterated integrals


compare? 4.10.21 Exercise. For the measure space (X, 5, f..l) that is the Fubinate of (Xl , 5 1 , f..ld and (X2 ' 52 , f..l2 ) , 5 contains the a-ring 5 consisting of all empty, gnite, or countable unions of sets of the form El X E2 , Ei E 5i , i = 1 , 2 . Is 5 necessarily 5?

[Hint: The case (Xi , 5i , f..li ) = ( [0, 1] , SA, >') , i = 1 , 2 , is relevant .]

4.10.22 Exercise. If

E = ffi? \ { (Xl , X2 ) XI - X2 E Q} and {Al , A2} C (5A2 n E) , then >. (A I X A2 ) = 0.

4.10.23 Exercise. Is there a signed measure space ( [0, 1] , 5, f..l) such that f..l =t=- 0, f..l « >., and for all a in [0, 1] , f..l( [0, aD = O? 4.10.24 Exercise. If f E MaN, then

f* : JR. '3 x H lim f(y) �f f(x-) ytx

exists and is in MaN and f* is left-continuous , i.e. , f* (x) = lim f* (y) ; furytx

thermore, f* � f. Similar results obtain for

f* (x) �f lim f(y) �f f(x+) . ytx

4.10.25 Exercise. If f E BV, then: a) U* , f* } c BV; b) there is a countable set 5 such that off 5, f* = f* ; c) for the jump function

j(x) �f L I f* (y) - f* (y) 1 y<x

associated with f, f - j E C(JR., q . [Hint: If f E MaN and f(x+) - f(x-) > 0, then

Q n (f(x-) , J(x+) ) :;to 0.]

00 4.10.26 Exercise. If 5 �f {an }NEN C JR., {jn }nEN C C, and L Un l < 00,

then f : JR. '3 x H L Un l i s in f E MaN and an'::;x

I f* (x) - f* (x) 1 = { Ijn l if x E � . ° otherwIse

n=l


4.10.27 Exercise. If (JR., S)" p,) is a complex measure space and for x in JR., f(x) �f p,[(-oo, x)] , then: a) f E BV; lim f(x) = 0; c)

xt-oo

f(x) = f(x-) [�f lim f(Y)] . ytx

d) The function f is continuous at x iff p,({x}) = O. (Properties a)-c) characterize functions of normalized bounded variation. The set of all such functions is NBV) .

4.10.28 Exercise. a) If f E JR.lR n NBV, then I : Co (JR., JR.) '3 9 f--t l 9 df is a DLS functional. The associated measure p, is totally finite and

clef [ ] Tf (x) = p, (-oo, x) .

b) If f E ClR n NBV, there is a corresponding complex measure p, and

Tf (x) �f 1 p, 1 [(-oo, x) ] .

4.10.29 Exercise. If f E BV( [a, b] ) , then varf i s continuous at c in (a , b) iff f is continuous at c.

-

4.10.30 Exercise. a) If 9 E BV([O, 1] ) , for some a and b,

g( [O, 1 ] ) C [a, b] .

b) If, for [a, b] as in a) , f' E C([a, b] , JR.) , then f o g E BV([O, 1] ) .

* * *

Littlewood's Three Principles

In closing this discussion of real analysis there is an opportunity to mention some general guidelines [Lit] that lie at the root of many of the arguments and ideas that have been presented.

Real analysis began with Newton in 1665. In his time, a function was usually given by a formula and most formulre represented functions that were (at worst) piecewise differentiable. As real analysis grew and developed over the succeeding 300 years, there appeared functions defined by expressions of the form

B if x E T { A if x E S

f(x) = � �; x E U '


The study of trigonometric series gave rise to highly discontinuous functions and led Cantor to discuss the sets of convergence and sets of divergence of the representing series . He turned his attention to set theory itself and started an investigation that climaxed in 1963 with P. J . Cohen's resolution of the Continuum Hypothesis [Coh] . Other significant outgrowths of Cantor's work were general topology, Lebesgue's theory of measure, DLS functionals , abstract measure theory, probability theory, ergodic theory, etc.

The subject of functional analysis arose in an attempt to unify the methods of ordinary and partial differential equations and integral equations. The techniques were approximations that permitted modeling the equations by systems of finitely many linear equations in finitely many unknowns. In passing from the solutions of the approximating systems to what were intended to be solutions of the original equations , limiting processes were employed. At this point there appeared the need to conclude that the functions found in the limit were within the region of acceptable solutions . Therein lies the virtue of the completeness of the function spaces LP and the condition that a Banach space be norm-complete. (H. Weyl remarked that the completeness of L2 is equivalent to the Fischer-Riesz Theorem. More generally, the norm-completeness of L1 (hence of LP) is essentially equivalent to the three basic theorems-Lebesgue's Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem-of integration.) Modern methods of TVSs relax the completeness condition somewhat-topological completeness replaces norm-completenessbut some effort is made to insure that limiting processes do not lead out of the spaces in which the solutions are sought .

Topology gave rather general expression to the notions of nearness (neighborhoods ) and continuity. Measure theory elaborated the notion of area since integration was for long viewed as the process of finding the area of a subset { (x, y) : 0 :::; y :::; f(x) } offfi? At the start , f was a continuous function and area was approximated by the areas of enclosed and enclosing rectilinear figures . The idea for this goes back first to Riemann, then Euler and Newton and ultimately Archimedes , who, using exhaustion, determined the areas and volumes of some nonrectilinear figures .

With the advent of Lebesgue's view of integration, rectangle took on a new meaning, in the first instance, the Cartesian product E X [a, b] of a measurable subset E of ffi. and it closed interval [a, b] .

Nevertheless , behind all the generalities lay the intuitive notion of the graph of a well-behaved function. When topology and measure theory were alloyed, the evidence became clear that many of the results were derivable by appeal to approximation of the general situations by others where the comfort of continuity and simplicity were available.

Paraphrased, Littlewood's Three Principles read as follows.


1. Every measurable subset of ffi. is nearly the union of finitely many intervals .

2. Every measurable function is nearly continuous. 3. Every convergent sequence of measurable functions is nearly uniformly

convergent . In the context of locally compact topological spaces and Borel measures

the statement corresponding to 1 . is : 1'. Every Borel set is nearly the finite union of compact sets .

4.10.31 Exercise. In what contexts is l' valid? 4. 10.32 Exercise. Which of the results in Chapters 1-4 exemplify Littlewood's Three Principles?

COMPLEX ANALYSIS

5 Locally Holomorphic Functions

5.1. Introduction

In ClR the sets Ck(JR., C) , k = 0, 1 , . . " satisfy the relations

Indeed, if fo = X [ ) and for k in N, 0,00

/k(x) �f { oLXoo /k-l (Y) dy if x 2': ° if x < °

then /k E Ck(JR., JR.) \ Ck+ 1 (JR., JR.) . Furthermore, for no x in JR. \ {a} is /k(x) representable by a power series of the form

00 L cn (x - a)n (�f P(a, x ) . n=O

On the other hand, as the developments in this Chapter show, if f E (C'c , r > 0, and

lim f(z + h) - f (z) �f I' (z) h->o h h>'O

exists for all z in D(O, rt �f { z : z E c, I z l < r } , the open disc of ra-00 dius r, then there is a power series L anzn �f P(a, z) such that for all

00 z in D(O, rt , f (z) = L cnzn, i .e. , the series in the right member con-

n=O verges if I z l < r and the sum is f(z) . Moreover, by abuse of notation, f E Coo (D(O, rt , q .

The striking contrast between the situations described in the two paragraphs above is one of many motivations for the study of CIC . The principal

203

204 Chapter 5. Locally Holomorphic Functions

tool used in the investigation is complex integration-a special form of integration, for which Chapters 2 and 4 provide a helpful basis. The principal results from those Chapters are the THEOREMs about functions "( in BV

and associated Riemann-Stieltjes integrals of the form 11 J d"(. In Section 1 .2, fl represents a special ordinal number. Henceforth,

unless the contrary is stated, the symbol fl, with or without affixes, is reserved for a region, i.e. , a nonempty connected open subset of C regarded as ffi.2 endowed with the Euclidean metric:

d : ffi.2 3 [(a , b) , (c, d)] H J(a - c) 2 + (b - d) 2 E [0, (0) .

Thus an element of C is regarded either as a pair (x, y) of real numbers or as a complex number z �f x + iy. When S e C, J E CS , and (x, y) E S, the notations J(x, y) and J(z) refer to the same complex number. Since polar coordinates r, fJ are frequently useful, when z = x + iy = reiO = a + reit , the notations J(r, fJ) and J (a + reit) are also used for J(x, y) and J(z) .

For a complex number z �f a + ib , the complex conjugate of z is a - ib and is denoted z . This use of - conflicts with - used for closure in topology. Henceforth, for a subset S of C, S denotes the set of complex conjugates of the elements of S, whereas SC denotes the closure of S regarded as a subset of C endowed with its Euclidean topology.

The term curve is reserved for a continuous map

"( : [0, 1] 3 t H "((t) E C;

the image of "( is "(* �f "(( [0, 1 ] ) . When ,,( E BV( [O, 1] ) , i.e. , when "( is rectifi

able, the integral t Jb(t)] d� (t) is sometimes' written 1 J dz, or 1 J dz, io "I "I'

or 11 J (a + re27l"it ) 27rire27l"it dt.

5.1 . 1 ExerCise. a) For n E Z+-) J(z) = zn , and "((t) = e21rit : 1 J dz = 0.

b) If J(z) = ! , then 1 J dz = 27ri . z "I Complex integration is used below to show that a function J differen-

tiable in a region fl is locally representable by a power series : If a E fl, then C contains a sequence {cn} nEN such that for some positive r(a) and all z for which I z - a l < r(a) ,

00 J(z) = L cn (z - a)n .

n=O

Section 5.2. Power Series 205

Thus the study of differentiable functions in CIC is reduced, in part, to the study, pursued with particular vigor and success by Weierstrafi, of power series convergent in some nonempty (open) discs of the form:

D(a, rr �f { z : z E C, I z - a l < r } . Later developments (v. Chapter 10) are concerned with methods of extending (analytically continuing ) , when possible, such functions to domains properly containing the disc within which the representing series converges .

5.2. Power Series

5.2. 1 THEOREM. (Cauchy-Hadamard) THE (FORMAL ) POWER SERIES

00 5 clef ,", n = � CnZ n=O

I 1 clef CONVERGES IF z = 0 OR I Z < --==..---1 = Rs ; 5 FAILS TO CONVERGE lim I cn l :;;: n ..... oo

(5 diverges ) IF I z l > Rs . 1 1 [ 5.2.2 Note. The conventions - = 00 and - = 0 are observed. o 00

Hence the radius of convergence Rs of 5 is in [0, 00] . For example, if Cn = n!, n E N, Rs = 0; if Cn = -\-' Rs = 00. It is occasionally

n. convenient to denote Rs by Rc to emphasize the dependence of the radius of convergence upon the sequence c �f {cn} nEZ+ of coefficients in 5.]

PROOF. If 0 < Rs :::; 00 and I z l < Rs , for some r in (0, 1 ) ,

_ 1 I z l lim I Cn i n < r, n ..... oo

whence, for some s in (r, 1 ) , all large N , and all k in N,

M S· 0 1 th t· 1 5 clef ,", n M 1').T C h mce < s < , e par la sums M = � Cnz , E n , are a auc y

n=O sequence.


If Rs < 00 and I z l = Rs + r5 > Rs , for some {nkhEN' nk t 00 and

(v. 1 .7.32) . o 5.2.3 Exercise. a) If Rs > 0 and S(k) is the power series derived from S by k-fold term-by-term differentiation, then RS(k) = Rs . b) For each S(k) , convergence is uniform on every compact subset of D (0, Rs t . c) If "I is a

00 rectifiable curve, "1* C D (0, Rs t, and f = L cnzn, then

n=O

[Hint: The Cauchy-Hadamard formula and induction apply for a) and b); b) implies c) . ]

5.2.4 Exercise. If "I is a rectifiable closed curve ["((0) = "1(1 ) ] , and n E N, then i zn dz = 0

5.2.5 THEOREM. IF "I IS A RECTIFIABLE CLOSED CURVE AND C tJ. "1* , FOR SOME n IN Z,

clef 1 1 1 Ind ,), (c) = -. -- dz = n. 27rl ')' z - C (5.2.6)

PROOF. Since c tJ. "1* , the integral in the right member of (5.2 .6) is welldefined. Furthermore, since ea = 1 iff (by abuse of notation) a E 27ri . Z (v. Section 2.3) , the result is valid iff for some n in N, exp [Ind ,), (c)] = exp(n) .

If E E (0, 1 ) , there is a piecewise linear "IP h;' is a polygon) such that

(v. 4.8.5 and 4.8.6) . Thus, since "I� is piecewise continuous (indeed, piecewise constant , whence "IP piecewise absolutely continuous) ,

clef [18 1 ] O(s) = exp ( ) d"lp (t) o "IP t - C

= exp [18 ( � "I� (t) dt] o "IP t - C

Of (s ) � O( s) "I� (s) "Ip (s ) - c

Section 5.2. Power Series 207

Hence (-yp(s ) - c)O' (s ) - (-yp (s) - c) ' O(s) == 0, i.e. , (_0_ ) ' == o. How"IP - C

o 0 ever, --- E AC ( [tk ' tk+d ) (v. 4.8.5) , whence --- is constant on "IP - c "IP - c each interval [tk ' tk+d . Thus, since __ 0- is continuous on [0, 1] ' for some "IP - c

o constant K, -- = K. "IP - c Consequently,

1 0(0) = K = 0(1 ) 0(1 ) "Ip (O) - c "Ip(O) - c "Ip(l ) - c ( ) = KO(I ) , "IP 0 - C

i.e., 0(1 ) = 1 {= exp [27riInd ,),p (c)] } and so for some mp in Z,

For some sequence ("(Pk hEI\P each like "IP , lim Ihpk - "1 1 1 00 = 0 and k-+oo

Hence, for some n in Z, Ind ')' (c) = n.

5.2.7 Exercise. The last sentence above is valid. [Hint:

For all appropriate P, the integrand in the first term of the right member above is small and f! ("IP) is bounded. The absolute value of difference of the approximating Riemann-Stieltjes sums for the second term is small.] [ 5.2.8 Remark. The discussion above suggests that for many purposes the hypothesis that "I E BV([O, 1] ) may be replaced by either of the (weaker) hypotheses

a) "I is piecewise linear; b) off a finite set , "I' exists and "I' is piecewise continuous on

[0, 1] .

o


Furthermore, each 11 f 0 'Y(t) d'Y(t) , i .e . , each f f dz, may be re

placed by 11 f 0 'Y(t) . 'Y' (t) dt . The last integral lends itself to the

methods of elementary calculus and avoids the use of RiemannStieltjes integration.]

[ 5.2.9 Note. If l' is a rectifiable closed curve, the function ind 'Y in 2.4.20 and Ind I' are the same, v. 5.4.28 and 5.4.37. The common value of ind 'Y (c) and Ind 'Y (c) is the winding number of l' with respect to c, v. 5. 2 .10. If l' is not rectifiable, Ind 'Y is not defined.]

5.2.10 Exercise. If 'Y(t) = e2mrit , n E Z, then ind 'Y(O) = Ind 'Y (O) = n.

5.2 .11 Exercise. a) If {'Yk }�=1 is a finite set of curves , there is one and K

only one unbounded component C of C \ U 'YZ . b) If c E C, k=1

K 2:)nd 'Yk (c) = o. k=1

K K

c) On each of the components of C \ U 'YZ , L Ind 'Yk (c) IS constant, v.

2.4.20c) . K

k= 1 k= 1

[Hint: a) Since U 'YZ is compact , for some positive r, k= 1

K U 'YZ ¥D (O, r) k=1

K and C \ D(O, r) is connected. b) L Ind 'Yk (c) is constant in each

k=1

component of ( C \ Ql 'YZ) . c)

. 1 1 hm -- dz = 0, 1 � k � K.] I c l-too 'Yk z - C

Section 5.3. Basic Holomorphy 209

The developments above set the stage for a general discussion of (locally) holomorphic Junctions , i .e. , functions that behave like many of those appearing above. They all enjoy the local property-differentiabilitywhich reveals itself as a crucial characterization of local holomorphy.

5.3. Basic Holomorphy

5.3.1 DEFINITION. WHEN U E O(C) , AN J IN CU IS holomorphic IN U (f E H (U) ) IFF I'(z) EXISTS FOR EACH z IN U. MORE GENERALLY, WHEN S e C, THE NOTATION H (S) SIGNIFIES THE SET OF FUNCTIONS HOLOMORPHIC IN SOME OPEN SET CONTAINING S:

{J E H (S) } {} {3U {{S C U E O(C) } 1\ {J E H (U)}}} .

The next two results show that despite the restrictiveness of the conditions defining holomorphy, if a E C and R > 0, H [D(a, Rr] is substantial.

00 5.3.2 Exercise. a) If the radius of convergence of L cnzn is R and, for z

n=O 00 in D(O, Rr, J(z) �f L Cnzn, then J E H [D(O, Rr] . b) If 0 < r < R and

n=O l'(t) = re27rit , 0 -::; t -::; 1 , then i J(z) dz = O.

[Hint: a) Induction yields the identity

wn - zn n- l n-2 n-3 n-2 --- - nz = (w - z) (w + 2zw + · · · + (n - 1 )z ) . w - z Thus when max { I z l , I z + h i } -::; r < R and I h l is positive,

I J(Z + h) - J(Z) � n- 1 1 h - L-t ncnz

n=l

= I h l l� Cn [(z + ht-2 + 2z(z + ht-3 + . . . + (n - 1) zn-2] I -::; I�I f n(n - 1 ) l en I rn-2 .

n=2 Then 5.2.3 applies . b) 5.2 .4 applies. ]

5.3.3 Exercise. If l' is a rectifiable closed curve, g E C (1'* , C) , and, for z not in 1'* , J (z) �f 1 g( w) dw, each component C of C \ 1'* is a region -y w - z and J E H (C) .


5.3.4 THEOREM. IF 1 E H (Q) ,

I(z) �f I(x + iy) �f u(x , y) + iv(x, y) [�f �(f) + iCS(f) ] ,

AND z E Q, THEN THE Cauchy-Riemann equations

ux (x, y) = Vy (x, y) and uy (x, y) = -vx (x, y)

OBTAIN. PROOF. If z = x + iy, then

!' ( z) = lim 1 ( z + h) - 1 ( z ) h�O h h";O

= Ux + ivx

= lim I(z + ih) - I(z) h�O ih h";O

= - iuy + vy .

[ 5.3.5 Remark. In terms of the operators

a �f � �f � (� _ i�) a �f � �f � (� + i�) az 2 ax ay ' az 2 ax ay

- al the Cauchy-Riemann equations reduce to a 1 = az = o.

In this book the same symbol a appears in different contexts: m au analysis , - , the partial derivative of u with respect to x; in topol-ax

ogy, a(U) , the (topological) boundary of U; in complex analysis , al as introduced above. Nevertheless, the intended meaning of a whenever it occurs, is clear.

Th . al al d · ·f h . 1 d · . e" notatlOn - resp. -= oes not slgm y t e partIa envatIve az az of 1 with respect z resp. z. The alternative notation a 1 resp. a 1 is less likely to be misinterpreted.]

o

5.3.6 Exercise. If: a) Q is a region in C; b) u and v are in ffi.r1 ; c) the derivative of the map T : Q 3 (x, y) r-+ [u(x, y) , v (x , y)] E ffi.2 exists ; d) in Q, the Cauchy-Riemann equations obtain, i.e. , Ux = Vy and uy = - Vx , then for some 1 in H(Q), I(x + iy) = u(x, y) + iv(x, y ) . (If partial differentiability of u and v only off a countable set and the Cauchy-Riemann equations only


a.e. in Q are assumed, a result of Looman and Menchoff implies nevertheless that J E H(Q) , v. [Sak] , (5. 1 1 ) , p .199. )

The next development leads to the connection between differentiability, i.e. , holomorphy , of a function J in en and, for each D(a, rt contained in Q, the represent ability of J (z) as a power series P( a, z) (converging at each point z of D(a, rt) . The fundamental tools are Cauchy 's Integral Theorems and Cauchy 's Jormul(£; .

5.3.7 Exercise. a) If {A, B, C} c e and a(ABC) is the 2-simplex determined by A, B, and C,

b) If 8[a(ABC)] = [A, B] U [B, C] U [C, A] . 1 (1 - 3t)A + 3tB

/,(t) = (2 - 3t)B + (3t - I )C

(3 - 3t)C + (3t - 2)A

. 1 If 0 < t < -- 3 'f 1 2 1 - < t < -3 - 3 '

'f 2 1 - < t < 1 3 - -

then /' E BV([O, 1 ] ) and /'* = 8[a(ABC)] . c) If J E C b* , C) ,

1 d f l 1 1 -J dz � + + J dz 'Y [A ,B] [B,C] [C,A]

�f r J dz Ja[a(ABC)]

1 2 = 13 J 0 /'(t) (3B - 3A) dt + 13 J 0 /,(t) (3C - 3B)

3

+ 11 J 0 /'(t) (3A - 3C) dt. 3

5.3.10 Exercise. If n E Z+ and J(z) �f z" ,

r J dz = o. Ja[a(ABC)]

[Hint: Since /" exists and is continuous on the intervals

the three integrals in (5 .3.8) , (5.3.9) can be calculated directly. ]

(5.3.8)

(5.3.9)


5.3.11 THEOREM. (Cauchy's Theorem, basic version) IF

[a(ABCW C Q

AND f E C(Q, q n H (Q \ {P}) , FOR "I AS IN 5.3.7, i f dz = o.

[ 5.3.12 Remark. The point P is not specified. The hypothesis f E H (Q \ {P})

(weaker than f E H (Q)) adds strength to the conclusion at the cost of complicating the proof. However, in the treatment of the elaborations of Cauchy's Theorem and in the treatment of Cauchy's formula, the strong(er) conclusion plays an important role.]

PROOF. For simplicity, the heart of the argument is carried out under the additional hypothesis that P tJ. [a(ABCW. In that circumstance, if J f dz �f K i- 0, then as in Figure 5.3. 1 , the barycenters (31 , (32 , (33 of

'Y [A, BJ , [B, CJ , [C, A] engender four simplices

Direct calculation reveals

c

A Figure 5.3. 1 .


and thus the absolute value of at least one of the summands in the right member above is at least I�I . For the corresponding simplex, the procedure just applied may be repeated and induction produces a sequence a( ABC) , al , . . . , an , . . . of subsimplices such that

a(ABC) ::) al ::) . . . ::) an ::) . . .

and I r 1 dZ I 2': I�I . Since �(an l 4

d· ( ) diam [a(ABC)] d fl [8 ( )] _ Q8[a(ABC)] } lam an < an 1: an - , - 2n 2n

for some Q in [a(ABCW, n (ant = Q. Since P tJ. [a(ABCW, P i- Q nEN

and f'(Q) exists. Hence, if Q i- z E (ant,

I(z ) - I(Q) - f' (Q)(z - Q) �f 8(z) , z - Q

and E > 0, then for some no , 8(z) < E if n > no . Hence, by virtue of 5.3.10, if n > no ,

I�I � I r 1 dz l = I r [J(Q) + f' (Q)(z - Q) + 8(z)(z - Q)] dz l 4 Ja(an l Ja(an l

= I r 8(z) (z - Q) dz l Ja(an l

� E diam [��ABC)] . f! [8 (an )]

diam [a(ABC)] f! {8 [a(ABC)] } < E · . ��--� - 2n 2n 11( 1 � E · diam [a(ABC)] . f! {8[a(ABC)] } .

Hence 1( = o. More generally, when P E [a(ABCW, there remain the following pos

sibilities : a) P E {A, B, C}; b) P E 8[a(ABC)] \ {A, B, C} ; c) P E a(ABC) . For a ) , if, e.g. , P = A, on (A, B ] there is a sequence {An} nEN such that

lim An = P (= A) . The original argument applies for a (AnBC) : n-+=

r 1 dz = 0, lim r 1 dz = r 1 dz. Ja[a(AnBCl] n-+= Ja[a(AnBCl] Ja[a (ABCl]


For b) , if, e.g., P E (A, B) , the argument for a) applies to cr(APC) and cr( P BC) . Furthermore,

r 1 dz = r + r 1 dz = 0 + o. Ja[a-(ABC)] Ja[a-(APC)] Ja[a-(PBC)]

For c) the argument for a) applies to cr(ABP) , cr(PBC) , and cr(PCA) , while r 1 dz = r + r + r 1 dz. 0

Ja[a-(ABC)] Ja[a-(ABP)] Ja[a-(PBC)] Ja[a-(PCA)]

5.3.13 Exercise. The PROOF above can be conducted via barycentric subdivisions of cr( ABC) .

5.3.14 Exercise. a) If 'Yp is a closed polygon, there are finitely many K K

simplices {crkh�k�K such that 'YP C U 8 (Sk) ' b) If U (crk/ c Q and k=1 k=1

1 E H (Q) , then 1 I(z) dz = O. "iP

5.3.15 THEOREM. IF Q IS convex AND 1 E C(Q, q n H (Q \ {P} ) , FOR SOME 1 IN H (Q) , 1 == (.1) ' . PROOF. If {Q, z} c Q, the convexity of Q implies [Q, z] c Q. The formula 1(z) �f r I(w) dw uniquely defines T Since Q is convex, if R E Q, then

J[Q,z]

[cr(QzRW C Q and 5.3. 11 implies 1(R) = r + r I(w) dw . Thus, J[Q,z] J[z,R] - - 1 [J(w) - I(z)] dw

if R i- z, I(R) - I(z) - I(z) = [z ,R] If E > 0 and R is R - z R - z near but different from z, owing to the continuity of I,

I r [J(w) - I(z)] dw l � E IR - z I . J[Z,R]

o

5.3.16 COROLLARY. IF Q IS convex, 'Y IS A RECTIFIABLE CLOSED CURVE, AND 'Y* C Q, FOR 1 AS IN 5.3. 15, J 1 dz = O.

"i

PROOF. If 'Y is piecewise linear, since (.1) ' = I, FTC implies

i 1 dz = 11 (.1) ' h(t)] 'Y' (t) dt

= 11 [10 'Y(t)] , dt = 1 0 'Y I � .


Since ,,/ is a closed curve J 0 "/ I �= O. If "/ is a rectifiable closed curve, as in the last part of the PROOF of

5.2.5, there is a sequence {"/Pn } nEN such that lim J 1 dz = J 1 dz. n-+= 'YPn "I

Each integral in the left member above is zero. 0 5.3. 17 Exercise. If Q is convex, {P, Q } c Q, and "/1 , "/2 are two rectifiable curves such that:

a) "/� U "/� C Q, b) "/1 ( 0) = "/2 (0) = P, "/1 ( 1 ) = "/2 (1 ) = Q,

then for any 1 in H (Q) , J 1 dz = J 1 dz: the integral is independent of "11 "12

the path. If, in 5.3.15, the hypothesis that Q is convex is dropped, J 1 dz can

"I be different from o.

clef { } ( 1 . 5.3 .18 Example. If Q = C \ 0 and 1 z) = - , then 1 E H (Q) , and Q IS Z

not convex. If "/(t) = e27rit , 0 :::; t :::; 1 , then "/* C Q and

On the other hand, since Ind 'Y (O) = 27ri, the presence of 27ri in the right member above suggests the possibility of a formula that relates Ind "I (0) and some integral involving I. In the derivation of the formula, the importance of the hypothesis 1 E C(Q, q n H (Q \ {P} ) in 5 .3 .11 becomes clear.

5.3.19 THEOREM. (Cauchy's integral formula, basic version) IF Q IS A CONVEX REGION, "/ IS A RECTIFIABLE CLOSED CURVE SUCH THAT "/* C Q AND 1 E H(Q) , FOR EACH a IN Q \ ,,/* ,

1 J I(z) I(a) · Ind 'Y (a) = -. -- dz. 27rl "I Z - a (5.3.20)

[ 5.3.21 Note. If Ind 'Y (a) = 1 and g(z) �f I(z)(z - a) , (5.3 .20) yields i I(z) dz = g(a) = 0, i.e. , a significant generalization of

5.3. 11 . The larger message of (5.3.20) is that the value of 1 at any point a in Q is a complex weighted average of the values of 1 on "/* . Mere continuity of 1 is insufficient for such a conclusion since, absent the assumption of the differentiabilty of I, 1 .2 .41 implies that for some continuous I, f(a)=l while 1 1"1' = 0.]


PROOF. Since J(a)Ind ,), (a) = � 1 J(a) dz it suffices to prove the valid-27rl ')' z - a

ity of 1 J(z) - J(a) dz = o. ')' z - a

To this end 5.3.16 is applicable because, by definition,

{ J(z) - J(a) (""\ clef Fa : H 3 z = z - a

f' (a) if z ¥= a if z = a

is in C(Q, C) n H (Q \ {a} ) and thus 5.3.16 implies (5.3.20) .

(5.3.22)

o

5.3.23 COROLLARY. IF a E Q, D(a, r) c Q, AND J E H (Q) , FOR SOME SEQUENCE {cn(a) }nEZ+ IN C AND ALL z IN D(a, rt ,

00 J(z) = L Cn (a)(z - a)n . (5.3.24)

n=O

PROOF. For "( : [0 , 1] 3 t f-t a + re27rit , Cauchy's formula and 5.2 .11 imply that if z E D(a, rt , then

J(z) = � (1 Fz (w) dw + 1 J(z) dW) = � (0 + 1 J(z)

dW) 27rl ')' ')' W - Z 27rl ')' W - Z

= _1 1 J(w) dw . 27ri ')' (w _ a) (1 _ z - a )

w - a

Since I z - a I < 1 for all w on "(* , 1

= f (� � :) n . When v is w - a 1 _ z - a n=O w - a

counting measure, Fubini's Theorem applied to

justifies the subsequent interchange of integration and summation:

Thus cn (a) = � 1 ( J(wj +1 dw . 27rl ')' W - a n

5.3.25 Exercise. If (X, S, f.-l) is a complex measure space,

g E 5, U E O(C) , and g(X) n U = 0,

o

Section 5.3. Basic Holomorphy

then f : U 3 z r-+ r �p,)(x) is in H (U) . ix g x - z

[Hint: If z E U, for some a in U and some positive r, z E D(a, rt c U.

I z - a I Iz - a l For each x in X, ( ) :::; -- < 1 and g x - z r

1 1 (Xl (z - a)n g(x) - a ( z - a ) = L (g(x) - a)n+l . (g(x) - a) 1 - n=O g(x) - a Again, when v is counting measure, Fubini's Theorem applies to

. J dp, (x) � J (z - a)n and valIdates ( ) = � ) )n 1 '] X g X - Z x (g(x - a + n=O 5.3.26 Exercise. In the context of 5.3 .23, if m E N, f(m) exists and

00 f(rn) (z) = L n(n - 1) · · · (n - m + l)cn (a)(z - a)n-rn

n=rn (whence f(rn) E H(Q) ) . In particular,

217

f(n) (a) 1 J f(w) Cn a = = - dw . ( ) n! 27ri 'Y (w - a)n+l (5 .3 . 27)

Furthermore the radii of convergence of the series representing f and f(rn) are the same. 5.3.28 Exercise. If M(a, r) �f max I f (w ) 1 �f max I f ( a + re27l"it ) I , Iw-a l=r O�t� l then M(a, r)2 z 1

1I f (a + re27l"it ) 1 2 dt and

00 = L i cn (a) 1 2 r2n . n=O

[Hint: Fubini's Theorem applies ; {e27l"nit } nEZ is ON on [0, 1] .]


The inequality 00 L I cn (a) 1 2 r2n � M(a, r)2 n=O

is Gutzmer 's coefficient estimate : which implies the (weaker) Cauchy esti-. If (

n

) ( ) 1 n!M(a, r) mate . a < . - rn

A function in H (C) is an entire function. The set of entire functions is denoted E.

5.3.29 Exercise. (Liouville) If f E E and sup I f(z) 1 � K < 00, then for zEIC

some constant c, f == c: a bounded entire function is a constant function. [Hint: Gutzmer's estimate applies for all positive r. Hence, if n > 0, then cn (O) = 0.]

5.3.30 Exercise. If f E E , a < 1, and sup I f (�)11 < 00, then f IS a zEIC 1 + z Q

constant . [Hint: Gutzmer 's estimate applies .]

n

5.3.31 Exercise. If n E N, p(z) == L akzk and an i- 0, then for some � , k=O

p(�) = 0 [the Fundamental Theorem of Algebra (FTA)] .

[Hint: If the assertion is false, g �f -7 E E and Liouville's Theorem applies. ]

5.3.32 COROLLARY. IF A IS A BANACH ALGEBRA, x E A, THEN sp(x) i- 0 (v. 3.5.20 ) .

PROOF. If sp(x) = 0 , then for all z , (x - ze) - 1 exists, and if x' E A� , direct calculation reveals that the function

f : C 3 z r-+ x' [(x - ze) - 1 ] E C

is entire. As I z l -+ 00, I f(z) 1 -+ o. Liouville's Theorem implies f == 0: for every x', (X-I , x') = O. Since X- I i- 0, the Hahn-Banach Theorem is contradicted. 0

If 0 i- z � x + iy, for some unique (J in [0, 271") , z = I z l eio �f reiO . If 00

f(z) �f L Cn zn

, Cn �f an+ibn , {an , bn } C JR., 0 � R < He , and z = ReiiJ , n=O


then 00 00

n=O n=O �f �[J(z)] + iCS[J(z)] �f u(x, y) + iv(x , y)

�f U(R, 0) + iV(R, 0) .

219

In these notations , the following items reveal some of the power of the basic version of Cauchy's formula. 5.3.33 Exercise.

1 1271" 1 1271" � (Co ) = - U(R, O) dO, CS (co ) = - V(R, O) dO, 27r 0 27r 0 1 1271" U(R 0) - inO dO .../.. Cn = -R ' e , n r o. 7r n 0

[Hint: If n E ;Z \ {O}, 1271" einO dO = 0.]

5.3.34 LEMMA. IF U(R, O) :::; M < 00 AND n E N, THEN

PROOF. If n -::J 0,

I cn l :::; 2 [M �� (co) ] . c

1 1271" 1 1271" -Cn = Rn [_U(R, O)e- inO] dO = Rn [M - U(R, O)e-inO] dO. 7r 0 7r 0

Because M - U(R, 0) � 0, 5.3.33 implies

l en I :::; 7r�n 1271" [M - U(R, O)] dO 2 [M - � (co )] and as R t Rc the right member above approaches . 0 R�

5.3.35 Exercise. If {In}nEN C H(Q) and for each compact subset K of Q, In I K� 1 1 K ' then I E H (Q) and for each k in N, I�k) I K� I(k) I K "

[Hint: The formula in (5.3. 27) applies .] The Maximum Modulus Theorem and the Open Mapping Theorem

discussed below are shown to be logically equivalent . The first is proved by appeal to the following fundamental principle:


The average of a finite set S of real numbers lies between max {} ) and min {S}. Thereupon the validity of both assertions is established.

5.3.36 THEOREM. (Maximum Modulus Theorem) IF I E H (Q) AND a E Q, THEN I I(a) 1 2': sup I I (z) 1 IFF I IS A CONSTANT FUNCTION: THE ABSOLUTE

zEn VALUE OF A NONCONSTANT FUNCTION, HOLOMORPHIC IN Q, CANNOT ACHIEVE A MAXIMUM VALUE IN Q. PROOF. If l'(t) �f a + re27rit , 0 :::; t :::; 1, Cauchy's formula and the averaging principle cited above imply

I I(a) 1 :::; 1 1 I I (a + re27rit ) I dt

:::; max I I (a + re27rit ) 1 = max I I(z) l . O�t� 1 zE8[D(a,r)O]

If max I I(z) 1 = I I (a) l , the argument for the Gutzmer estimate zE8[D(a,r)o] 00

yields L Icn (a) 1 2 r2n :::; II(aW = Ico (a) 1 2 , whence cn (a) = 0, n E N, i .e. , I n=O

is a constant function. Hence, if I is not a constant function, at every interior point a of Q,

I I(a) 1 is not a local maximum value of I I I : I I(a) 1 < sup I I (z) 1 0 zEn

5.3.37 Exercise. (Minimum Modulus Theorem) If I Eo H (Q) , 0 tJ. I(Q) and for some a in Q, I I(a) 1 2': inf { 1 1(z) 1 : z E Q }, then I is a constant .

[Hint: The function g �f -7 is in H (Q) .]

5.3.38 COROLLARY. IF Q IS BOUNDED , I E H (Q ) , AND FOR EVERY a IN 8(Q) , inf sup II (z ) 1 = M, THEN sup I I(z) 1 :::; M.

N(a)EN(a) zEN(a)nn zEn

PROOF. Only if M < 00 and I is not a constant is an argument required. For those circumstances, there is a sequence {K m} mEN of compact subsets such that Km C K�+1 ' m E N, and each compact subset K in Q is contained in some Km, (v. 4.10.9) . Moreover 5.3.36 implies that

max I I(z) 1 �f Mm zEK=

is achieved at some point Zm on 8 (Km) . For all m, Mm :::; Mm+1 . Since Q is bounded, {Zm } mEN contains a convergent subsequence, again denoted {zm}mEN" If lim Zm �f Zoo , since Km C K�+1 ' m E N, Zoo E 8(Q) . The m-too


hypothesis implies that if E > 0 and m is large, Mrn < M + E. Hence, for any z in Q, I f(z) 1 :::; M + E. 0

5.3.39 THEOREM. (Open Mapping Theorem) IF f E H (Q) , f(Q) IS EITHER A REGION OR A SINGLE POINT, i.e. , f(Q) IS A NONEMPTY REGION OR f IS A CONSTANT FUNCTION.

PROOF. If !' =j=. 0, f' i- 0 in some nonempty open subset V of Q. If a E V, then for some positive r , D( a, r t c V, and if 0 :::; fJ :::; 27r, then

f (a + reiO) - f(a) I -'---------:-°0:---- = f (a) + Er e · r& '

For small positive r, IET,o l < � 1 !, (a) 1 and I !, (a) + ET,o l > � 1 !, (a ) 1 �f 15.

Hence, if r is small, min I f (a + reiO ) - f(a) 1 > rr5 �f 2d. The geome°SOS2". try of the situation reveals that if w E D(f (a) , dt ,

If g �f w - f and 0 tJ. g (D(a, rn, 5.3.37 implies

i.e., d > lw - f(a) l 2': min { lw - f (a + reiIJ) 1 : 0 :::; fJ :::; 27r } > d, a contradiction. Hence, for some b in D( a, rt, w = f(b) ,

D(f(a) , dt c f (D(a, rn 0

[ 5.3.40 Note. The PROOF above of the Open Mapping Theorem resorted to the Minimum Modulus Theorem, a consequence of the Maximum Modulus Theorem:

{Maximum Modulus Theorem} ::::} {Open Mapping Theorem} .

Conversely, the Open Mapping Theorem implies that a nonconstant function f in H (Q) maps Q onto a region: if a E Q, for some positive r, the open set f (D(a, rt) is contained in f(Q) and for some positive s , D(f(a) , s ) C f (D(a, rn. If f(a) = 0, each f(b) on 8 [D(f(a) , stJ satisfies I f(b) 1 > If(a) 1 = O. If f(a) i- 0, the half-line starting at 0 and passing through f ( a) goes on to meet 8 [D(f(a) , srJ in a point f(b) such that I f (b) 1 > If(a) l : {Open Mapping Theorem} ::::} {Maximum Modulus Theorem} .J


5.3.41 THEOREM. (Inverse Function Theorem) IF J E H (Q) , a E Q, AND J'(a) i- 0, FOR SOME POSITIVE r , J I D(a,r) O IS INJECTIVE AND J- 1 1 J[D(a,r) O ] IS HOLOMORPHIC.

PROOF. Since J'(a) i- 0, for all z in some D(a, rt, J'(z) i- O. The function { J(w) - J(z) if z i- w G : Q x Q 3 (z, w) r-+ w - z I'(z) if z = w

which is in C (Q2 \ { (z, w) : z = w }) is shown next to be in C (Q x Q, q . Since I' E H (Q) , I' is continuous . If z is near but different from w ,

and l'(t) �f ( 1 - t )z + tw , then

1'(0) = z, 1'(1 ) = w, 1" (t) = w - z, 1 1 1

G(z, w) - G(z, z) = w _ z 0 {Jh(t) ] }' dt - G(z , z) ,

= 11 [I'h(t)] - I'(z)] dt,

whence IG(z, w) - G(z, z) 1 is small: G E C (Q x Q, q . Thus, for some positive s , s < r and if max{ l z - a i , Iw - a l } < s , then

IG(z, w ) 1 � � I J'(a) 1 > 0,

1 i.e. , if {P, Q} C D(a, s ) , then I J(P) - J(Q) I � 2 1 1' (a) I ' IP - Q I . Hence J ID(a.s) O is injective: for some g defined on J (D(a, s t) , g o J(z) == z .

If b E Q1 and Q1 3 w i- b, for some P, Q in D(a, st , g(w) - g(?) P - Q

w - b J(P) - J(Q) " (5.3.42)

Since Q E D(a , st , J'(Q) i- o. As w -+ b, P -+ Q and the right member of 1 (5.3.42) converges to I'(Q) ' 0

[ 5.3.43 Note. If 0 = a = J(a) , 00

w = J(z) = LanZn , 0 :::; I z l < r, a1 i- 0, n= 1

and when Iw l is small, 00

z = g(w) = L bmwm . (5.3.44) m=1


Thus

Comparison of coefficients leads to a sequence

of formulre from which the sequence {bn} nEN is (recursively) calculable in terms of the sequence {an} nEN . (Similar but more complicated formulre obtain when a and I(a) are more general. ) The Cauchy estimates imply for M �f M(a, r ) ,

Loo M n M 2 1 If l z l < r, the series l a1 I z - -z = l a1 I z - -2 z --z repre-

n=2 rn r 1 - -r sents a function F, a majorization of I, i.e. , a function F for which the power series coefficients majorize in absolute value the power series coefficients for I. The equation W = F(z) is quadratic in z, and if

then G o F(z) = z. At and near 0,

whence, for some positive p depending only on 1 1' (0) 1 r and M, G' exists in D(O, pt . There is a sequence {cn}nEN such that for

00 W in D(O, pt , G(W) = L Cn Wn (= z) . The recursive formulre

n=1 for the sequence {cn}nEN show I bn l :::; I cn l , n E N. Thus the series (5.3.44) converges if Iw l < p:

1 (D(O, rn ::J D(O, pt ·J

223

5.3 .45 Example. If I(z) = eZ , then 1 E E , I' = I, and f' is never zero. Although 1 is locally injective it is not globally injective : for n in Z, e2mri = 1 .


5.3.46 Exercise. If 0 < r < l a l , for some L in H (D (a , rr ) , eL (z) = z. [Hint: The function L in 2.4.21 serves .] The condition !, (a) i- 0 plays a central role in the PROOF of the major

part of the Open Mapping Theorem, 5.3.39. There is a refinement that deals with the circumstance: 1 is not a constant function, but !, (a) = O.

5.3.47 THEOREM. IF 1 E H(Q) , a IS IN Q, AND 1 IS NOT A CONSTANT FUNCTION, FOR SOME m IN N, SOME NEIGHBORHOOD N(a) , A g IN H [N(a)] ' AND ALL z IN N(a) I(z) = I(a) + [g(z)]Tn . FURTHERMORE, FOR SOME b AND A POSITIVE r , g[N(a)] = D (b, rr �f V, g I N(a) ' IS INJECTIVE, AND FOR SOME h IN H(V) , h 0 g I N(a) (z) == z.

PROOF. For some m in N and some N(a), if z E N(a), then

I(z) = I(a) + (z - a)Tn [� cn (z - a)n 1 �f I(a) + (z - a)Tnk(z)

k' k' and k I N(a) i- O. Hence k E H [N(a)] ' and for some h , h' = k ' If z E N(a) , then {k(z) . exp[-h(z)] }' = exp[-h] [k' - kh'] = O. Hence, for some con-def '0 stant M = IMl e" , k exp (-h) = M on N(a) . Since

k = [ IMI ,!; exp ( h :iO) ] Tn ,

if g(z) �f (z - a)k(z) , then I(z) = I(a) + [g (z) ]Tn. Moreover, g(a) = 0 and g'(a) = k(a) i- o. 0

5.3.48 THEOREM. (Morera) IF U E O(q , a E U, 1 E C(U, q , AND FOR EACH 2-SIMPLEX a CONTAINED IN U \ {a} , r 1 dz = 0, THEN 1 E H (U) .

Ja(a) [ 5.3.49 Note. The open set U need not be connected, e.g. , the conclusion is valid if U �f D(O, lrl.,JD(3, lr �f l!Jl.,JD(3, lr .]

PROOF. If r > 0 and w E D(b, rr c U, the hypothesis implies (even when b = a) that the formula F( w) � r 1 dz unambiguously defines an F in

J [b,wl cD(b,r)o . Moreover, FTC implies F' exists throughout D(b, rr and F' = I. Hence F E H (D(b, rn , and thus F' (= f) E H (D(a, rr ) · 0 5.3.50 Example. The hypothesis 1 E C(U, q in 5.3.48 cannot be omitted. Indeed, if

1 ( z) = { 0:2 if z i- 0 otherwise


and a is 2-simplex contained in C \ {O}, then r I dz = o. Nevertheless lara) there is no entire function g such that g l lC\{o} = I.

Another aspect of holomorphy is highlighted by contrast in

5.3.51 Example. For each k in N, the function 1 ( ( 27r) ) k SIn -

j" ll!. " X h

0

2;' if x i- 0

otherwise

is continuous on JR., Ik (�) = 0, k, n E N, and if k i- l , h and II are different functions:

Two (different ! ) functions in C(JR., JR.) can assume the same values on an infinite set, e.g. , S �f { .!. } , such that S· i- 0.

n nEN The following result, which is of general importance in the context of

holomorphic functions, provides the contrast to 5.3.51.

5.3.52 THEOREM. (Identity Theorem) IF

I E H(Q) , S �f {adkEN C Q, and Q ::) S· i- 0,

THE VALUE OF I(z) IS DETERMINED FOR ALL z IN Q BY THE VALUES {I (ak ) }kEN · PROOF. By hypothesis , for some a in S· and some subsequence {an } nEN contained in Q, lim an = a. If g E H (Q) and g (an ) = I (an ) , n E N, for n-+= some positive r, I - g �f h is representable by a power series in D( a, r t and h (an ) = 0, n E N. If h i- 0, for some M in N and all z in D(a , rt ,

and Co i- o. But since all but finitely many an are in D( a, r) 0 , for all

large n, 0 = h (an ) = (an - a)M (�o Crn (an - a)rn) . Since an - a i- 0,

= for all large n, Co + L Crn (an - a) rn = o. The second term in the left

rn=1


member above converges to zero as an ---> a, whence Co = 0, a contradiction: h ID(a,r) o = 0

If Q = D( a, r t the argument is complete. If b E Q \ D( a, r t there is a set { [zn , zn+dh�n�N of complex intervals such that Z1 = a, ZN = b, and

N P �f U [zn , zn+d c Q (v. 1.7. 11 ) . If w E P, for some positive r(w) , h

n= 1 is representable for all Z in D[w, r(w)]O by a power series . For each r, the open cover {D[w, r(w) ] O} wEP of the compact set P, admits a corresponding finite subcover: {D (wi , rit L<i<r These may numbered so that in the passage from a to b along P, w� (= a) , W2 , . . . , WI (= b) are encountered in succession. Since P is connected,

The argument of the preceding paragraph implies h lD( . . ) 0 = 0, 1 � i � I. Wz , Tz

[ 5.3.53 Remark. Loosely paraphrased, the Identity Theorem says that the behavior of a function f holomorphic in region Q is determined by the behavior of f on any infinite subset S such that S· c Q.]

The next result is an application of 5.3.52.

o

5.3.54 THEOREM. IF Q �f C \ {O} , FOR NO f IN H(Q) IS THE EQUATION exp[f(z)] = Z VALID (THROUGHOUT Q) .

def · 0 PROOF. If f = u + iv and Z = e' , 0 � () � 271", then u (z) = 0 and for some k : '][' f-> Z, V (eiO ) = () + 2k (eiO ) 71". Since v and () are continuous, k is a constant. As () r 271", v (eiO ) ---> v(l) while () + 2k71" ---> (}(1) + (2k + 1)71" =I- v(l ) , a contradiction. Thus f is constant on '][' and 5.3.52 implies f i s constant on Q. Since u(2) =I- 0 = u (l ) a contradiction emerges. 0

A region Q is star-shaped if for some a in Q, Q is the union of half-open complex intervals [a , b) .

5.3.55 Exercise. A convex region Q is star-shaped; the cOnverse is false. 5.3.56 Exercise. If Q is star-shaped and 0 tJ. Q then H(Q) contains an f such that exp[J(z)] = z.

[Hint: For some a in Q, f : Q 3 Z f-> r dw serves.] J[a,zl w

5.3.57 Exercise. If f �f u + iv E H(Q) , then

dcl dcl l1u = Uxx + Uyy = l1v = Vxx + Vyy = o.


[Hint: Near each point a + ib of Q, f is representable by a convergent power series . Hence near (a , b) , both u(x, y) and v (x, y) are representable by convergent power series of the form

00

Tn ,n=O

Thus u and v are infinitely differentiable. The Cauchy-Riemann equations apply.]

[ 5.3.58 Note. The expression t:m �f Uxx + Uyy is the Laplacian of u. In terms of the operators a and a t:m = 4aau. As introduced, l1 is applied to functions u in C2 (Q, JR.) ; nevertheless , l1 is applicable to functions u in en so long &'i the partial derivatives Uxx and Uyy exist . If u E en and in Q, l1u = 0, then u is harmonic and the set of functions harmonic in Q is, in analogy with H (Q) , Ha (Q) . Owing to the linearity of the operator l1, {f �f u + iv E Ha (Q) } {} { {u, v} C Ha(Q) n JR.n �f HaJH:(Q) } ,

H (Q) C Ha(Q) . Hence the study of harmonic functions can be and is confined to Rvalued functions .

On the other hand, because the real and imaginary parts of holomorphic functions are harmonic, there is a strong connection between the theory of functions in H (Q) and the theory of functions in HaJH: (Q). The relationship of the two theories is discussed in Chapter 6.]

227

The set HaJH: (Q) consists of the JR.-valued functions that are harmonic in Q. When u E HaJH: (Q) and for some v in HaJH: (Q) ,

f : Q 3 (x, y) f-t u(x, y) + iv(x, y) �f f(x + iy) is in H (Q) , then v is a harmonic conjugate (in Q) of u. If v is a harmonic conjugate of u and c is a constant, v + c is also a harmonic conjugate of u.

1 5.3.59 Example. The equation In I z l = - In I z l 2 and direct calculation re-2

veal that u : e \ {o} �f Q 3 (x , y) f-t In I z l is in HaJH: (Q) . However, if

f �f u + iv E H (Q) , i.e., i f a global harmonic conjugate v of u exists, for z in Q,

1 exp[J( z ) ] 1 = 14


If I z l = 1, for some k(z) in Z, I (z) = 2k(z)7ri . As in the PROOF of 5.3.54, I is constant , whereas �[J(z)l (= In I z l ) is not.

Global harmonic conjugates need not exist.

5.3.60 THEOREM. (Vitali) IF:

a) In E H (Q) , n E N AND sup I l ln l l oo :::; M < 00; nEN b) {akhEN C Q AND lim ak �f ao E Q \ {adkEN ; k-+oo c) lim In (ak ) EXISTS FOR ALL k IN N; n-+oo

FOR SOME I IN H (Q) AND EVERY COMPACT SUBSET K IN Q,

PROOF. If K is a compact subset of Q, for some positive r ,

U D(a, 3rt C Q. aEK If {a, b} c K and 1 b - al < r, Cauchy's formula implies

I ln (a) - In (b) 1 :::; M

11 I b � al dz l = 2M l b - al

27r I z-al=2r r r

{In}nEN is equicontinuous . The Arzela-Ascoli Theorem (1 .6.9) shows that {In}nEN contains a subsequence converging uniformly on K, i .e. , {In }nEN is a normal family.

There are compact sets Krn such that Krn C K;;'+I , m E N, and each compact subset K in Q is contained in some Krn , (v. 4.10.9) . Hence there are subsequences {Inrn} EN such that: a) n ,Tn 1"l

b) as n -+ 00, {Inrn} converges uniformly on Krn . Hence there is a function I such that if K is a compact subset of Q, then Inn l K� 1 1 K . In particular, for each n, I E H (K� ) , i .e. , I E H (Q) . In sum:

The sequence {Inn} nEN derived from the diagonalization method just applied converges uniformly on every compact subset of Q. If, for some a in Q, {In (a) } nEN fails to converge, owing to the hy

pothesis a) , {In } nEN contains two subsequences {I�i) } nEN ' i = 1 , 2 , con-

vergent at a and such that Al �f lim 1�1 ) (a) i- lim 1�2) (a) �f A2 . The n---+CXJ n---+CXJ


diagonalization technique applied to each of these subsequences yields subsubsequences {f�� } resp. {f�� } that converge uniformly on each

nEN nEN compact subset of Q to functions f( l ) resp. f(2) in H (Q) . For k in N, f�� (ak ) = f�� (ak ) , whence f( l ) (ak ) = /2) (ak ) . Furthermore, 5.3.52 im-plies f( l ) = f(2) and {I( I ) } u {1(2) } C {fn } . Hence nn nEN nn nEN nEN

f( l l (a) = lim f�� (a) (= Ad n-+=

f(2) (a) = lim f�� (a) (= A2) . n-+=

The leftmost members above are equal, whereas Al -j. A2 , a contradiction.

[ 5.3.61 Note. The terminology locally uniformly convergent is used to describe a net of functions converging on each compact subset of an open set U.

Thus 5.3.60 may be paraphrased as follows.

A sequence of functions holomorphic, uniformly

bounded in a region Q, and convergent on a subset having a limit point in Q is locally uniformly convergent in Q.

Similarly, a locally uniform Cauchy net resp. a locally uniformly bounded net is a net that is uniformly a Cauchy net on every compact subset of U resp. a uniformly bounded net on every compact subset of U. These notions and their specializations are of particular utility in complex analysis , e.g. , in the application of the Arzela-Ascoli Principle in conformal mapping, the Great Picard Theorem, etc. The precise ways in which the applications occur are described in the course of the remainder of the book.]

o

5.3.62 Exercise. If f E H(Q) and 1R(f) == 0 resp. CS(f) == 0 on Q, then f is an imaginary resp. real constant on Q.

[Hint: The Open Mapping Theorem applies. ]

230

5.4. Singularities

5.4.1 Example. If

Chapter 5. Locally Holomorphic Functions

f z) = { Z if z ¥= ° ( 6 otherwise '

then as defined, f E H(C \ {O} ) , f is not continuous at 0; f tJ. H (C) . However, since f(z) = z on C \ {a}, f(O) can be redefined to be 0, and then the newly defined f is in H (C) .

If if z ¥= °

otherwise then g E H (C \ {O} ) . Since I g (z) 1 is large when I z l is small and positive ( I g( z) 1 is unbounded near 0 ) , there is no way to define g at ° so that the resulting function is in H (C) .

If

then:

h(Z) = { �± if z ¥= ° otherwise '

a) if z is positive and near 0, Ih(z) 1 is large; b) if z is negative and near 0, Ih(z) 1 is small; c) if z E iffi. and z ¥= 0, 1 h( z) 1 = 1 , whence there is no entire function E

such that EIIC\{o} = h. The phenomena illustrated above motivate the following

5.4.2 DEFINITION. WHEN fl IS A REGION, s E fl, AND f E H(fl \ {s} ) , s IS AN isolated singularity OF f. 5.4.3 THEOREM. IF s IS AN ISOLATED SINGULARITY OF f, EXACTLY ONE OF THE FOLLOWING OBTAINS:

a) FOR SOME POSITIVE R, IF 0 < r < R, THEN I f(z ) 1 IS UNBOUNDED IN D(s, rt \ {s} : s IS A pole OF f;

b) I f(z) 1 IS BOUNDED IN SOME NEIGHBORHOOD N(s) : s IS A removable singularity OF f, i.e. , f IS DEFINABLE AT s SO THAT THE RESULTING FUNCTION IS HOLOMORPHIC IN N(s) ;

c) (WeierstraB-Casorati) IF N(s) C fl, THEN f[N(s) \ {s}] IS DENSE IN C: s IS AN isolated essential singularity OF f.

PROOF. If c) fails, for some N(s) contained in fl, some a in C, and some positive r, f(N(s) \ {s} ) fails to meet some D(a, rt, F � _1_ is bounded f - a near s, and is holomorphic in D( s, rt \ {s} .

Section 5.4. Singularities 231

If G(z) �f (Z - S)2 F(z) , then G E H [N( s) \ {s }] . Direct calculation shows G'(s) exists and G(s) = G' (s) = 0. Thus near s, and for some N in

00 N \ {I} , G(z) = (z - s )N L cn (z - s)n and Co i- 0. If

n=O

n=O 1 then ¢ E H [N(s)] . For all z in some N1 (s) , ¢(z) i- ° and - E H [N1 (s)] ,

1 ¢

i.e., for z near s, ¢( z) is representable by a power series :

1 ex:> -(z) = L dn (z - s )n , do i- 0. ¢ n=O

1 ex:> Hence near s, f(z) = a + )N-2 L dn (z - s)n . If N - 2 = 0, then f (z - s n=O is definable at s so that f E H [N (s)] : a) obtains.

If N - 2 > 0, for I z - s I positive and small, I f (z) I is large: b) obtains. o

Further discussion of holomorphy and singularities is simplified by the following notations and terminology, when U is an open subset of C.

• When f E H (U) , Z(f) �f { a : a E U, f(a) = o } . Each a in Z(f) is a zero of f and Z(f) is the set of zeros of f·

• When S c U, U \ S is open and f E H (U \ S), then:

P(f) <!..._ef { a ·. S d · 1 f f } a E an a IS a po e 0

(the set of poles of I) and

E(f) �f { a : a E S and a is an isolated essential singularity of f } .

(the set of essential singularities of I) . When S = P(f) , then f i s meromorphic in U (even when S = 0) . The

set of functions meromorphic in a region Q is M(Q) .

5.4.4 Exercise. If f E M(Q) , then P(ft n Q = 0. [Hint: If a E P(ft n Q, a is not an isolated singularity.] [ 5.4.5 Note. The number N - 2 in the PROOF of 5.4.3 is the order of the pole s.


Not all singularities are isolated. For example, if

1 if z -j. 0 and sin - -j. 0 z otherwise

and if, for some N(O) , k E H [N(O)] , for all sufficiently large n in

N, k E C (D (0, n�) \ D (0, 2�7r ) 0) . Since

clef ( 1 ) ( 1 ) ° Kn = D 0, - \ D 0, - E K(C) n7r 2n7r

clef { 1 } S = z : z = - , n E !Z \ {O} c P(k) , 2n7r and a contradiction emerges . Zero is a limit point of poles of k; zero is a nonisolated singularity of k. In Chapter 10 the treatment of analytic continuation leads to the general definition of a singularity (singular point) of a function. The more detailed behavior of functions near isolated essential singularities is treated in Chapter 9.]

5.4.6 Exercise. If J E H (Q) and a E Z(f) , for some least no in N, there is a sequence {cn (a)} �=o such that co( a) -j. 0 and for all z near a,

00 J(z) = (z - a)no L cn (a)(z - a)n : n=O

a is a zero oj order or multiplicity no. 5.4.7 Exercise. If Q \ S is a region, J E H (Q \ S) , and a E P(f) n Q, then for some least no in N there is a sequence {cn (a)}�=o such that co (a) -j. 0

1 ex:> and for all z near a, J(z) = ( ) '" cn (a) (z - at: a is a pole oj order or multiplicity no.

z - a no � n=O 5.4.8 Exercise. If U is open and J is a non constant function in M (U), then {Z(ft U [P(fW} n U = 0.

[Hint: 5.3.52 and 5.4.4 apply. ]


5.4.9 THEOREM. IF 1 E H (Q) , AND a IS A ZERO OF ORDER no , THEN FOR SOME NEIGHBORHOOD N(a) , WHENEVER b E I(N(a) \ {a} ) ,

i.e. , 1 IS AN no-to- 1 MAP OF N (a) \ {a} . PROOF. For some positive r, if z E D(a , rr �f N1 (a) , then

00 I(z ) = (z - a)no L cn (z - at �f (z - a)nog (z)

n=O ,

and if z E N1 (a) , g(z ) i- O. Thus !L E H [N1 (a) ] ' and since N1 (a) is convex, g 1 g'(() g' if, for z in N1 (a) , G(z) = -

( - de then G E H [N1 (a)] and G' = - .

[a,z] g () g (Jl...- ) ' _ eGg' - g (eG) ' eGg' - geGG' Thus G - 2G 2G = 0 and for some constant

K, g = :K+G . It foU:ws that if h(z) =e

(z _ a) exp ( K � G) , then

K+G(a) Since h'(a) = e-n-o - i- 0, the argument in 5.3.41 shows that h is injective on some open subneighborhood N2 (a) of N1 ( a) and that h [N2 (a)] is open. Hence, for some t in (0, 1 ) ,

I f 0 i- b E D (0, tnO t, for some s and some fJ, clef { clef ...L [ (fJ + 2k7r) ] } If S = Wk = S no exp i nO

: 0 :::; k :::; no - 1 , then

and # ({wo , . . . , wno-d) = no. Since h is injective on N2(a) , h is also injective on the (open) neighborhood

Hence N ( a) \ {a} contains precisely no points Zo , . . . , zno - 1 such that


D

5.4.10 Exercise. If I E H (Q) and for some b, a is a zero of order no of 1 - b, for some function ¢ and some neighborhood N(a) : a) ¢ E H [N(a)] ; b) 1 = b + ¢no ; c) ¢ is injective and ¢' is never zero in N(a). 5.4. 11 COROLLARY. a) IF a E Q, I E H (Q \ {a} ) , AND a IS A POLE OF ORDER no OF I, FOR SOME POSITIVE R,

{ I b l > R} :::;. {# [J- l (b) n Q] = no} . b) FOR SOME POSITIVE r AND AN INJECTIVE ¢ IN H [D( a, r) ° ] , I = ¢nu . PROOF. a) If g(z) �f (z - a)no /(z) , 5.3.41 implies that for some positive g(z) . s, g is injective on D(a, sr · Then I I D(a,s) O ' which is z H (z _ a)no ' IS an

no-fold map of D(a, s)O on { z : I z l > � �f R } . b) 5.4.10 applies. D

5.4.12 Exercise. If I E H (Q) and l in is injective, f' ln is never zero and 1- 1 E H [/(Q) ] .


5.4.13 Exercise. If I(z) = eZ , then: a) for each a in C, 1'(0') -j. 0; b) for some positive r , I I D(<>,r) O is injective; c) for the g in the PROOF of 5.4.11 ,

g l f(D(<>,r) o ) E H (Qd; d) g (Qd c U Ln (w) ; e ) g'(w) = �; f) i f h E H (Q) , wEn, a E Q, and h( a) -j. 0, for some positive r, D( a, r r c Q and for some L in

H [D(a, rr] ' h(z) I D(a,r) o = eL(z) . If r > 0 and I E H [D(a, rr] ' the basic version of Cauchy's integral

formula (5.3.20) leads to (5.3.24 ) , i .e . , the representation of I by a power series converging to I throughout D( a, r r . A global version of Cauchy's formula leads to a similar representation when a is an isolated singularity of I. If a is not a removable singularity, the representation of I cannot converge at a but at best in the region D( a, r r \ {a} . A generalization of this kind of region is an annulus , i .e . , when 0 :::; r < R < 00, a region of the form { z : r < Iz - a l < R } �f A( a; r, Rr (for which the closure is A( a; r, R) �f { z : r :::; Iz - al :::; R } ) . An annulus A( a; 0, R) resp. open annulus A( a; 0, Rr is a punctured disc resp. a punctured open disc at a and is denoted D( a, R) resp. D( a, Rr . (For a given annulus A( a; r, R) when 0 :::; fJ < ¢ :::; 27r the open annular sector is

A(a : r, R; fJ, ¢r �f { Z : z = a + pei,p , r < p < R, fJ < '1/J < ¢ } . )


A reasonable approach to such a representation for an J holomorphic in A( a; r, Rr, involves, when r < s < S < R, an integration over the two curves, 'Yl and 'Y2 such that

(The set 'Y� U 'Y� is the boundary

8 [A(a; s, srl = { z : Iz - a l = s } u { z : Iz - a l = S } . ) For any w not in A( a; r, Rr, direct calculation shows that

Ind 'Y! (w) + Ind 'Yz (w) = o.

These remarks motivate the following

5.4.14 THEOREM. (Cauchy's integral formula, global version) IF U IS AN OPEN SUBSET OF C, J E H (U) , {-yd l:s;k«,K ARE RECTIFIABLE CLOSED

K CURVES , U 'YZ C U, AND FOR EACH w NOT IN U, k=l

K FOR EACH a IN U \ U 'YZ , k=l

K 2:)nd 'Yk (w) = 0, k=l

[ K 1 1 K J J( ) J(a) 2:)nd 'Yk (a) = -. L � dz. 2�l � - a k= l k= l 'Yk

(5.4 .15)

PROOF. Because the left member of (5.4. 15) i s z:.-valued and depends con

tinuously on w, V �f { w : t Ind 'Yk (w) = o } is open, contains C \ U, k=l and so V U U = C.

For the function

G : U x U '3 (z, w) H w _ z if z -j. w { J(w) - J(z) I'(z) if z = w


introduced in the PROOF of the Inverse Function Theorem, (5.3.41 ) , the hypothesis (5 .4 .15) implies that the formulre

1 K L J G(z, w) dw

G(z) dgf t Z �(w� du k= i 'Yk

if z E U

if z E V

are consistent on U n V: G is well-defined throughout C. Since G E H (U) , C E H (V) , and U n V =I- 0, i t follows that C E E . Because

K V ::) n Ind 'Yki (O) , k=i

for some positive R, V ::) { z : I z l > R }. For large I z l , IC(z) 1 is small and thus 5.3.29 implies C = 0. The promised conclusion follows when the equation C(z) == ° is written in terms of the defining formulre for C. 0

[ 5.4.16 Note. The hypothesis (5.4. 15) is satisfied if, e.g. , as in Figure 5.4 .1 , for some positive R,

U = D(O, Rr , {adi�k�K C D(O, Rr , K

'Yk (t) = ak + rke27rit , t E [0 , 1] , 1 :::; k :::; K, and U 'YZ C D(O, Rr · k=i In Figure 5.4 .1 , the dashed lines together with the small circles themselves may be construed as the image of a single rectifiable

Figure 5.4.1

Section 5 .4. Singularities

closed curve, say r. Integration over r can be performed so that the integrations over each dashed line are performed twice (once in each direction) with the net effect that those integrations con-tribute nothing to 1. The validity of 5.4.14 for the configuration just described follows directly from the basic version of Cauchy's Theorem. The approach in 5.4.14 permits a very general result free from appeals to geometric intuition, v. Figure 5.4.l .]

K 5.4.17 Exercise. In the context above, L 1 J dz = o.

k=1 'Yk K

[Hint: If a E U \ U 'YZ and 1(z) �f (z - a)J(z) Cauchy's formula k= 1

applies to T]

237

5.4.18 Exercise. If bk}�=1 and {Jj };'= l are two sets of rectifiable closed K J

curves such that U 'YZ U U J; c U and for each w not in U, k=1 j=1

J K L Ind 8; (w) = L 1nd 'Yk (w) , j=1 k=1

J K then L 1 J dz = L 1 J dz. j= 1 8; k= 1 'Yk

[Hint: 5.4. 17 applies to the calculation of J K

Ll fdz - Ll J dz.] j=1 8) k=1 'Yk

The following is a useful consequence of 5.4.14.

5.4.19 THEOREM. IF J E H [A(a; r, Rt] ' FOR SOME {cn (a) }nEZ IN C AND ALL z IN A(a; r, Rt ,

00 (5.4.20)

n=-(X) THE RIGHT MEMBER OF (5.4 .20) , THE Laurent series FOR J IN THE ANNULUS A( a; r, Rt, CONVERGES UNIFORMLY ON EACH COMPACT SUBSET OF A( a; r, R) 0 •


PROOF. If Z E A( a; r, Rr, for some [s , S] contained in (r, R) ,

Z E A(a; s, Sr.

If 'Yl (t) = a + se27ri( 1 -t) and 1'2 (t) = a + Se27rit , 5.4.14 implies

J(z) = � [1 J(w) dw + 1 J(w) dW] . 27rl W - Z W - Z ')'2 ')'1

For w in the first resp. second integral of (5.4 .21 ) ,

Hence 1

w - z resp.

1 w - z

I w - a I < 1 resp. I � I < 1 . z - a w - a

1 1 � (w - a)n - -z---a . -----:-w,,----a-:- = - � (z - a)n+ l 1 - -- n=O z - a 1 1 � (z - a)n

w - a 1 _ z - a - � (w - a)n+l . n=O w - a If s < t < T < S, both series converge uniformly in A( a; t , T) . If

{ -� 1 J (w ) (w - a)n dw if n :::; -1 ( ) 27rl ')'1 Cn a = 1 J J(w) - dw if n 2': 0

27ri ')'2 (w - a)n+ l

(5.4.21 )

(5.4.20) obtains. D [ 5.4.22 Remark. Owing to the fact that J is not assumed to be holomorphic in some neighborhood of a, none of the coefficients cn (a) , in particular those for which n < 0, need be zero.]

5.4.23 Exercise. If, for all z near but not equal to a, 00

n=-(X) and (v. 5.4.3)

a) Cn = 0 when n < 0, then a is a removable singularity of J; b) for some negative no ,

if n = no if n < no '


a is a pole of order -no; c) if inf { n : n E ;2;, Cn i- O } = -00, a is an isolated essential singularity

of f. 00

5.4.24 DEFINITION. WHEN f (z) = L cn (z - a)n IS VALID FOR ALL z n=-(X)

NEAR BUT NOT EQU AL TO a, THE residue of f at a IS c 1 �f Res a (f) . WHEN a IS A POLE OF f, THE principal part of f at a IS

- I Pa(f) �f L cn (z - a)n n=-(X)

(A SUM INVOLVING ONLY FINITELY MANY TERMS! ) . 5.4.25 Exercise. If a is a pole of order one of f, then

Res a (f) = Fm (z - a)f(z) .

5.4.26 Exercise. If no < 0 and a is a pole of order -no of f, then

. 1 d-no- I [(z - a)-no f(z)] Res a (f) = !1E1 -:-(_-n-_-1:-C) ! dz-no- I z of:- a 0

5.4.27 Exercise. Res a [Pa(f)] = Res a (f ) . 5.4.28 THEOREM. (Residue Theorem) IF f E M (Q) AND hdlSkSK IS A

K SET OF RECTIFIABLE CLOSED CURVES, S �f U 'YZ c [Q \ P(f)] , AND FOR

k= 1 EACH w NOT IN Q,

THEN

K L Ind T'k (w) = 0, k= 1

2�i t 1 f dz = L Res a (f) · [t 1nd T'k (a)] . k= 1 T'k aEP(f) k= 1

(5 .4 .29)

(5.4.30)

PROOF. The set S is compact ; hence, for some positive r, S ¥D(O, r) . If

F � { a : a E P(f) , t, Ind T'k (a) i- 0 } is unbounded,

F n [C \ D(O, r)] i- 0,


K and 5.2 .11b) implies that if l a l is large L Ind ,),k (a) = 0, a contradiction

k=l of the definition of F: F is unbounded.

If U is a component of C \ S and U is unbounded, for each b in U, K L Ind ')'k (b) = O. If b E 8(Q) , then b E UC, whence the continuity of the k=l

K K Z-valued function L Ind ')'k implies L Ind ')'k (b) = o. Hence, if F, which

k=l k=l is bounded, is not finite, then Fe i- 0.

K K If b E Fe , then L Ind ')'k (b) i- 0, because L Ind ')'k is a continuous Z-

k=l k=l valued function. Since 1 E M(Q) , Fe n Q = 0 (v. 5.4.8) , whence b E 8(Q) , a contradiction. In sum,

F = 0 or F is a finite set, say F = {al , . . . , am} . Thus the sum in the right member of (5.4.30) contains at most finitely

many nonzero terms. If F = 0, the global Cauchy Theorem implies both members of (5.4.30)

are zero. m If F = {ai , . . . , am} , then h �f 1 - L P ai (f) has only removable sin-

i= l K

gularities in Q \ (P(f) \ F) and thus L J h dz = 0 and 5.4.14 applies. k= l ')'k

l' 5.4.31 Exercise. If U E 0(((:) and 1 E M (U) , then f E M (U) .

[Hint: If a E U, for all ,z near a , and for some nonzero C- I , the l' Laurent series for f takes the form

if a rf. [Z (f) U P(f)]

otherwise .]

o

5.4.32 Example. a) If n E ;2;, 1 (z) = (z - a) n , and "( is a rectifiable closed 1 J l' curve such that a rf. "(

*, then -. -I dz = n · Ind ,), (a ) . If Ind ,), (a) = 1, 27rl ')'

the formula above may be interpreted as a means of calculating { the order of a - I x (the order of a)

if n > 0 [a E Z (f)] if n < 0 [a E P(f)] .


b) If {m, n} C ;2;+ , I(z) = (z - a)Tn + ( 1 ) , ,,( is a rectifiable closed z - b n

curve such that { a, b} r:t. "(*

, and 0 < 1 z - bl < 1 a - b l , then: bI) the Laurent series for 1 takes the form

b2) the coefficients {dp } pEZ in the Laurent series for j can be calculated by comparison of coefficients of like powers of z - b in the two members of

Tn (z _

-b�n+l + L kCk (Z - b)k-l k= l = Cz � b)n + � Ck (Z - b) k) . Ct

oo dp (z - b)P) .

A similar calculation when 0 < I z - a l < la - bl provides the Laurent series

f eq(z - a)q for j . The Residue Theorem 5.4.28 implies q=- oo

If Ind ')' ( a) = Ind ')' (b) = 1 the formula above may be interpreted as calculating the order of a (a zero of I) minus the order of b (a pole of f) .

For a function 1 in M(Q) and an a in Z(f) U P(f) , Ord f (a) denotes the order of a (as a zero or a pole of I ) . If Ord f ( a) = 1 , a is a simple zero or pole of I. The preceding formulre have the following generalization.

5.4.33 THEOREM. IF 1 E M(Q) , "( IS A RECTIFIABLE CLOSED CURVE SUCH THAT "(* c { [Q n D(O, rrl \ [Z(f) U P(f)] } , AND Ind ,), (w) = 0 WHENEVER W tJ- [Q n D(O, rr] ' THEN

- - dz = " • 1 J f' 27ri 1 � ')' aE[Z (f)nD(O,r)D]

bE [P(f)nD(O,r)D ]

[ 5.4.34 Remark. For the map ¢ of 2 .4.18,

¢ { [/b(t)] } E Arg {/b(t)] } .


The formula in 5.4.33 may be used to calculate the total change in 4> {fb(t)] } as "(

* is traversed. Owing to 2.4.18, the change is a multiple of 271" and is independent of the choice of the map 4>. The formula is known as the Principle of the Argument or the Argument Principle . It is most useful when

For example, if Q = C and "((t) = re2rrit , 0 ':::; t .:::; 1 , the formula provides the difference between the sum of the orders of the zeros of f in D(O, r t and the sum of the orders of the poles of f in D(O, rt · A detailed discussion i s given in 5.4.37.]

PROOF. At each point a in Z(f) resp. b in P(f) , the Laurent series for the meromorphic function j takes the form

The Residue Theorem 5.4.28 applies. D

5.4.35 Exercise. If f E H (Q) , a E Q, and f(a) i- 0, for some positive r, D(a, rt c Q, ° tJ. f (D(a, rt) �f QI , and for some L in H (Qd then eLof = f.and (L 0 I)' = j . Furthermore, if "( is a rectifiable curve and "(

* C D(a, rt, then L 0 f 0 "( E BV([O , 1] ) and

1 f' dz = 11 d[L 0 f 0 "( (t) ] (Riemann-Stieltjes integral ! ) . 'Y f °


5.4.36 Exercise. Under the hypotheses of 5.4.33, if g E H (Q) , then

1 1 f' - g- dz = " 271"i f � 'Y aE(Z(J)nD(O,r) O ) g( a)Ord f( a )Ind 'Y ( a)

g (b )Ord f (b )Ind 'Y (b) . bE (p(J)nD(O,r ) O )

[Hint: The Laurent series for gj at the points in Z(f) and P(f) are useful.]


5.4.37 THEOREM. (Argument Principle) IF : a) I E M(Q) ; b) "( IS A RECTIFIABLE CLOSED CURVE; AND c) "(

* c {Q \ [Z(f) u p(fm, THEN:

I clef A ) 0 "( = r IS A RECTIFIABLE CLOSED CURVE;

B)

ind r (O) = aE [Z(J)nfl] bE [p(J)nfl]

[ 5.4.38 Remark. The left member of the formula in b) is the winding number of the curve 1 0 "( about O. The result is most useful when, for each c in Z(f) U P(f) , Ind ,, (c) = 1 .]

PROOF. a) Since 1 1' 1 is bounded on "(*

, if 0 .:s; r < s .:s; 1 , for some con

stant M, I r(s) - r (r) 1 = 1 1 I' (z) dz l ::; M h(s) - "((r) l , v. 4.8.6. h(r) ,,,(s)]

Because "( is rectifiable, r is rectifiable; because "( is closed, r is closed. b) By virtue of 5.4.35 and the compactness of "(

*, for some Riemann

partition {tkL�k�n of [0, 1] and positive numbers {rd l�k�n : bl)

n "(

* c U D b (tk ) , rkt ; k= l

b2) D b (tk ) , rkt n D b (tk+l ) , rk+lt -j. 0, 1 .:s; k .:s; n - 1 ; b3) there is an £k in H {D b (tk ) , rkt } and such that e£k oj = I; b4) £k+ l 0 1 - £k 0 I is constant on D b (tk ) , rkt n D b (tk+d , rk+lt , and in 27ri . Z. Finally, 5.4.28 applies. D

5.4.39 COROLLARY. (Hurwitz) IF {In }nEN C H(Q) , 0 tJ- U In (Q) , AND nEN

In � I ON EACH COMPACT SUBSET OF Q, EITHER 1 == 0 OR 0 tJ- I(Q) . PROOF. If I t o, since I E H (Q) , Z(ft n Q = 0. Hence, if a E Q, for some positive r, 0 tJ- J [ A( a; 0, r r] and j E H [A (a; 0, � fJ . Furthermore,

1 1 f 1 1 1' lim -. fn dz = -. -

I dz. The left member above is

n--+= 27rl I z-al=� n 27rl I z-a l=� zero, whence so i s the right .

If I t O, for each a in Q, I is not zero in some neighborhood of a: o tJ- I(Q). D


5.4.40 THEOREM. (RoucM) IF {f, g} c H (Q) , D(a, r) c Q, AND

(5.4.41 )

THE SUMS OF THE ORDERS OF THE ZEROS OF f AND OF f + g IN D( a, r t ARE THE SAME:

aEZ(f)nD(a,r)O aEZ(f+g)nD(a,1') O

PROOF. For t in [0, 1] ' the hypothesis (5.4.41 ) implies that the integral

-1- 1 (f + tg)' dz �f N(t) 27ri I z-al=r f + tg

is well-defined. According to 5.4.33, N(t) is Z-valued. On the other hand, the left member above is a continuous function of t and must be a constant . Moreover,

N(O) = aEZ(f)nD(a,r)O

N(l) = aEZ(f+g)nD(a,r)O

and, since N is a constant function, N(O) = N(l) . D

5.4.42 Exercise. If, for the polynomials p, q, deg(p) = M < N = deg(q) and R is sufficiently large, then { I z l 2: R} :::;.. { lp(z) 1 < I q(z) I } · 5.4.43 Exercise. If aN i- 0,

N- l f( ) clef "" n ( ) clef N --I- 0 d h z = � anz , g z = aN z , aN r , an = f + g, n=O

5.4.40 and 5.4.42 imply that for R sufficiently large,

L Ord h (a) = N, aEZ(h)nD(a,R)O

i.e. , the strong form of FTA is valid: if p is a polynomial of degree N and multiplicities of zeros are taken into account , p has N zeros. 5.4.44 Exercise. a) If a is a simple pole of f, in some non empty open N( a) \ {a} , f is injective. b) If a is a simple pole of both f and g, some linear combination h �f of + (3g is holomorphic in some nonempty open neighborhood of a.

Section 5.5. Homotopy, Homology, and Holomorphy 245

5.5 . Homotopy, Homology, and Holomorphy

The close connection between ind 'Y and Ind 'Y when "( is a rectifiable closed curve suggests that there is a topological basis for many of the results about complex integration. An approach that reveals this basis is found in the next paragraphs. The fundamental material about homotopy is given in Section 1 .4.

5.5.1 DEFINITION. A CLOSED CURVE "( : [0, 1] H Y IS null homotopic in A IFF FOR SOME CONSTANT CURVE 15 : [0 , 1] '3 t H r5(t) == y E Y: "( AND 15 ARE homotopic in A.

5.5.2 DEFINITION. FOR TWO CURVES "( AND 15 IN A TOPOLOGICAL SPACE Y, WHEN "((I) = 15(0) THE product "(15 IS THE CURVE

{ "((2t) "(15 : [0, 1] '3 t H

r5 (2t - 1 )

1 IF ° < t < -- - 2 1 IF - < t < 1 2 -

The end "((I) of "(* is the start 15(0) of 15* . The curve-image ("(15)* : a) connects "((0) to 15(1 ) ; b) is the union of the two curve-images, "(* and 15* .

If 1] : [0, 1] H Y is a curve such that 1] (0) = Yo (a curve starting at Yo) there is an associated curve 1]- 1 : [0 , 1] '3 t H 1](1 - t) starting at Y1 �f 1](1 ) .

5.5.3 Exercise. The product ( �f 1]1]- 1 i s a loop starting at Yo and ( is null homotopic loop. 5.5.4 Exercise. a) The product operation for loops starting at Yo induces a binary operation on the set 7r1 (Y, Yo) of homotopy equivalence classes of all loops starting at Yo. With respect to this binary operation and its associated inverse, 7r1 (Y, Yo ) is a group, the fundamental group of Y. The identity of 7r1 (Y, Yo) is denoted 1 . b) The fundamental group is independent of the choice of Yo , whence may be denoted simply 7r1 (Y) .

[Hint: If Y1 E Y and a is a curve connecting Yo to Y1 , then

[ 5.5.5 Note. When, as in Section 1 .4, F provides a homotopy, the set F( · , S) SE [O, l ] may be viewed as a one-parameter family of graphs in [0, 1] x Y. Then ( 1 .4 .2) expresses the circumstance that the graph of "( is continuously deformed into the graph of 15.

In what follows Y is some region Q while "( and 15 are curves such that "(* U 15* c Q; as s traverses [0 , 1] ' the curves F( · , s) constitute


a family that begins with "( and ends with 15. Furthermore, for each s , F( · , s ) * e n.

Unless the contrary is stated, when "( and 15 are closed curves, i .e. , loops, and "( "'F,rI 15, the condition

F(O, s) == F(l , s) (5.5.6)

is imposed: each curve F(·, s) is assumed to be a loop, F is a loop homotopy in n, and F( · , 0) and F(· , 1 ) are loop homotopic in n.]

5.5.7 Example. Two loops can be homotopic in a region n via an F that is a homotopy in n but is not a loop homotopy in n. If n �f C \ { � } , "((t) = 3e27rit , r5(t) = 2e47rit , and F(t, s) = (3 _ s)e27r( I+S)it :

5 . 1 • F(t, s) -j. "2 (otherwIse, s = "2 and

a contradiction; • "( "'F,rI 15;

5 2 '

• both "( and 15 are loops, although if 0 < s < 1 , the curve F( · , s) is not a loop: F(O, s) = (3 - s) -j. F(l , s) = (3 - s)e27r( l+s)i .

5.5.8 DEFINITION. A REGION n IS simply connected IFF EACH LOOP "( SUCH THAT "(* e n IS LOOP HOMOTOPIC IN n TO A CONSTANT MAP.

5.5.9 Exercise. If n is simply connected, "( is a loop such that "(* e n, a E n, and r5 (t) == a, for some F, "( "'F,rI 15 .

5.5.10 Exercise. A convex region is simply connected.

5.5.11 Exercise. A star-shaped region is simply connected.

5.5.12 LEMMA. IF: a) "( AND 15 ARE LOOPS SUCH THAT "(* U 15* e n; b) VIA SOME LOOP HOMOTOPY F, "( "'F,rI 15; AND c) C IS A COMPONENT OF U �f C \ F ( [0, 1] 2 ) , THEN ind F(. , s) (a ) IS CONSTANT AS s TRAVERSES [0, 1] AND a REMAINS IN C.

PROOF. Owing to 2.4.20, ind Fe,s) (a) is defined and is a continuous ;2;valued function on the connected set [0, 1] x C. D

Section 5.5. Homotopy, Homology, and Holomorphy

[ 5.5.13 Note. Although F in 5.5.4 is not a loop homotopy in n of "I and 15, via some G : [0, 1] 2 H C, "I and 15 might be loop homotopic in n. However, the inequality

ind � (�) = 1 # 0 = ind 8 (�) and 5.5.9 deny the possibility.]

5.5.14 LEMMA. IF "I AND 15 ARE RECTIFIABLE LOOPS, a E C , AND

247

Ih - 15 1 1 00 < 1 1 "1 - a l l oo , (5 .5 .15)

THEN Ind � (a) = Ind 8 (a ) . PROOF. (A pictorialization of (5 .5 .15) provides an intuitive argument for

. clef 15 - a the conclusion. ) Owing to (5.5. 15) , a tJ- ("!* u 15* ) , whence, If 1] = -- , "I - a then

1 1 1 - 1] 1 100 < 1 , i.e. , 1]* C D(I , It : 1] i s a loop contained in D(I , lr and so Ind 1J (O) = O . On the other hand,

0 = Ind 1J (O) = '!L dt = -s: - dt - -"I - dt = Ind 8 (a) - Ind 8 (a) . 11 , 11 15' 11 ' o 1] 0 u - a 0 "I - a D

5.5. 16 Exercise. a) If "I is a rectifiable curve, "1* e n, and E > 0, for some polygon 7r* and some homotopy F, "I rv F,n 7r and

sup h(t) - 7r(t) 1 < E. tE [O, ! ] b) If "I and 15 are rectifiable curves, "1* U 15* c n, and for some homotopy

F, "I rv F,n 15, there is a finite sequence {7rZ L <k <K of polygons and a finite sequence {FrnL:S;k:S:K+ I of homotopies such that

and Ih(t) - 7rI (t) l l oo < E, I I 7rl (t) - 7r2 (t) 1 1 00 < E,


c) If Q is simply connected, "I is a rectifiable loop such that "1* C Q, and J E H (Q) , then i J dz = o. (In particular, the result applies when

clef 1 a tJ- Q and J(z) = -- : Ind /, (0' ) = 0.) z - o'

d) If J E H(Q) , "11 and "12 are rectifiable curves such that

"I� U "I� C Q, "II (0) = "12 (0) , "11 (1 ) = "12 (1 ) , and "11 "'n "12 ,

then 1 J dz = 1 J dz. /'1 /'2

e) If J E H(Q) , "11 and "12 are rectifiable loops such that

then 1 J(z) dz = 1 J(z) dz. /'1 Z - a /'2 z - a

[Hint: 5.5.11 .]

5.5.17 Exercise. If Q is simply connected and J E H(Q) , for some F in H(Q) , F' = J.

[Hint: If {a, z} C Q and "I is a rectifiable curve such that "1* C Q, "1 (0) = a, "1(1 ) = z, the value of F(z) �f i J dz is independent of

the choice of "I.]

5.5.18 Exercise. If Q is simply connected, J E H(Q) , and 0 tJ- J(Q) , for some g in H (Q) , J = eg o

f' [Hint: For some F, F' = j.]

5.5.19 Exercise. If Q is simply connected, J E H (Q) , 0 tJ- J(Q) , and n E N, for some g in H (Q) , J = gn .

[Hint: 5.5.18.]

If "I : [0, 1] H Q is a rectifiable loop, then ;Y : H(Q) '3 J H i J dz is a

linear functional defined on the vector space H(Q) . The set of all finite �r clef � - f h 1· f . 1 · b 1· Z sums = "11 + . . . + "In 0 suc mear unctlOna s IS an a e Ian group n.

If a E Q and "I (t) == a h is a constant map) , then ;Y �f 0, the identity of Zn.

If a tJ- Q the function Ja : Q '3 z H �_1_ is in H (Q) . 27rl z - a

In Z� there is a subset-::-actually a subgroup-Bn consisting of all r such that for all a not in Q, r (fa ) = o. The quotient group

Section 5.5. Homotopy, Homology, and Holomorphy 249

is the first homology group of Q over z. Sums r and l1 are homologous in Q (r R::n l1) iff r - l1 E Sn . All constant maps are pairwise homologous. When ;Y E Zn, "(* is a cycle in Q. When ;:y E Sn , "(* is a bounding cycle in Q.

5.5.20 Exercise. a) The groups Zn,�

Sn, and HI (Q--2 Z) are uniquely determined by Q alone. b) If F E H (Q) , Q = F(Q) , and r E Zn, there is a (�) d f - -corresponding (J r � F ("(I ) + . . . + F ("(n ) in Zn. The map

is a group homomorphism. c) if 0 is the identity of Zn ' (J (Sn) = O. d) If r - H E Sn , then (J (r) = (J (H) : (J is a homomorphism of HI (Q, Z) into

HI (fl, z) . [Hint: d) If (3 tJ- fl, then r � = 1 (F(z) - (3)' dz. The

iFh) W - (3 'Y F(z) - (3 Argument Principle applies.]

5.5.21 Exercise. If Q is simply connected, HI (Q , Z) = {O} .

5.5.22 Exercise. If Q � A(O; 1, 2r, then HI (Q, Z) = Z.

[ 5.5.23 Note. The definition given of HI (Q, Z) is tailored to the developments of this book. There are connect ions between the approach above and De Rham's Theorem [SiT] . More general definitions of homology groups are given in [Sp] .]

5.5.24 Exercise. If "( is a loop such that "(* C Q and "( IS null loop homotopic in Q (5.5.1 ) , then ;:y E Sn, i .e . , ;:y R::n O.

Loop homotopic loops are homologous.

5.5.25 Example. The sets L� �f IDEFGBCDI, L; �f IAHGFCBAI in Figure 5.5.1 (p. 248) correspond to loops LI , L2 , neither of which, nor their product L3 �f LI L2 , is null loop�omotopic in Q �f C \ ({p} u {q} ) . On the other hand, although neither LI nor L2 i s null homologous, L3 is null homologous.


H G F E

p

A \;:--....I...------'---J D B C

I Figure 5.5. 1 .

Homologous loops need not be loop homotopic.

5.5.26 Exercise. If "/* C Q, for some m in N, {"/} contains a closed polygon 7r* such that 7r* C Q and each side of 7r* is a vertical or horizontal complex interval of length not exceeding 2-Tn .

[Hint: If "/ E b} E 7rl (Q) , for some m, the distance between any point of "/* and any point of C \ Q is at least 2-m . For some N 1 in N, if I t - t' l < N ' then h (t) - ,,/ (t' ) 1 < TTn. The polygon 7r*

consisting of the segments ["/ (�) , ,,/ ( k � 1 ) ] , 0 :::; k ::;; N - 1 , lies in Q and 7r is homotopic to "/.] Discussions of the fundamental group 7r1 (Q) can be conducted in terms of rectangular polygons described above.

5.5.27 Exercise. For a set ,,/1 , . . . , "/n of rectifiable loops in Q and a E Q: a) there are rectifiable curves CI , . . . , Cn such that the product

is the homotopy equivalence class of a loop beginning and ending at a; b) if each "/i begins and ends at a, hi · "/2 · . . . . "/nf= 'h + . . . + 'h i c) if {a, ,6} C b}, then a = (3; d) h : 7r1 (Q, a) '3 b} H ;;;jBn E HI (Q, Z) is a well-defined surjective homomorphism; e) if {"/} in 7r1 (Q, a) is the product of commutators, i .e . , if

Section 5.6. The Riemann Sphere

then h ( h } ) = o.

[ 5.5.28 Note. Although HI (10) � 7r1 (10) � Z (v . [Arm, Lef]

and 5.9.23), for some Q, e.g., C \ ({ I } U { I} ) , 7r1 (Q) is not abelian (v. 10.5. 13) . On the other hand, for the commutator subgroup C of 7r J (Q ) , HJ (Q, Z) � 7rI (Q)/C: HI (Q, Z) is the abelianization of 7r1 (Q) [Arm, Lef] .]

5.6. The Riemann Sphere

The Riemann sphere is the compact subset

L:2 �f { (�, 1], () : (�, 1], () E ffi.3 , e + 1]2 + (( �) 2

= � } ( = a { B [ (0, 0, �) , � ) ] } of ffi.3 . For the injection (stereographic projection)

G : L:2 \ { (O, O, I ) } '3 P �f (�, 1], () f---+ G(P) �f z E C, illustrated in Figure 5.6. 1 ,

l:2

e (P) Figure 5.6. 1 . The Riemann sphere and stereographic projection.

251


there is defined the function f : L? 3 P H f 0 8(P) . Behavior of f(z) for large I z l is the same as behavior of f(P) for P near (0, 0, 1 ) or alternatively as behavior of f ( �) for z near and different from 0.

Similarity of L [N08(P)] and L(NQP) implies that

8[(� , 1], ()] = 1 � ( + i 1 = ( : 8 is a homeomorphism. Furthermore :[2 itself may be viewed as (the homeomorphic image of) the one-point compactification Coo �f O:J{ oo} of C. In that context, 8 has a unique extension e to :[2 and e : :[2 H Coo is a homeomorphism. Regions, curves, etc. , in Coo are 8-images of regions, curves, etc. , on :[2 .

When Q is a region in C, then 8(Q) is by definition a subset of C. When Q is viewed as a subset of Coo , 00 may well be a boundary point of Q. Thus

if Q is bounded otherwise

When {a, b} c Coo , J (a , b) is the Euclidean distance between 8- 1 (a) and 8- 1 (b) (points in :[2 , a subset of ffi.3 ) .

5.6.1 Exercise. The function J i s a valid metric in Coo . In the context described, a neighborhood of 00 in C is equivalently

described as a set containing, for some positive R, { z : I z l > R } or, for some positive r , as the image under 8 of { p : P E :[2 , 0 < J(P, N) < r } (a punctured neighborhood of (0, 0, 1 ) on :[2 ) . A neighborhood of 00 in Coo is for some positive r the set { z : J (8- 1 (z) , N) < r } or equivalently, for some positive R, { z : I z l > R } .

The behavior of f for large I z l is that of f(P) for P near but not equal to N. Thus the locutions f has a pole at 00, f has an essential singularity at 00, etc. , may be construed equally well as descriptions of the behavior of f (�) for z near but not equal to zero or of the behavior of f(P) for P near but not equal to N or of the behavior of f(z) for values of z in Coo and near but not equal to 00.

5.6.2 Exercise. a) If f is a polynomial of positive degree, f has ,a pole at 00. b) If f is entire and is not a polynomial, f has an essential singularity at 00. c) If 00 is a removable singularity of an entire function f, then f is a constant . 5.6.3 Exercise. If f (z ) = eZ , a -j. 0, and R > 0, for some z such that Iz l > R, eZ = a.

[ 5.6.4 Note. The result above and illustrates the two famous theorems of Picard. In summary form, they say that if a is an isolated

Section 5.6. The Riemann Sphere

essential singularity of I in Coo and for the punctured neighborhood N(a) \ {a} , e.g. , A(a; 0, R) , # {C \ I [N(a) \ {a}] } > 1 , then I is a constant .

In the neighborhood of an isolated essential singularity the range or image of a nonconstant function omits no more than one complex number.

Picard's Theorems (v. Chapter 9) are substantial strengthenings of the Weierstrafi-Casorati Theorem, 5.4.3c) . On the other hand, FTA says that the range of a nonconstant polynomial, a special kind of entire function having only 00 as a pole, omits no complex number.]

253

5.6.5 Exercise. If a is a pole of I and N (a) is a neighborhood of a, for some positive R, I(N(a) \ {a}) ::) { z : I z i > R } : for z near but not equal to a pole of I, I(z) omits no complex number of large absolute value.

5.6.6 Exercise. The Maximum Modulus Theorem (5.3.36) may be reformulated for Coo as follows .

If Q is a region in C, I E H (Q) and for each a in 800 (Q) and some M, inf sup I /(z) l :::; M, for all z in Q, I /(z) 1 :::; M.

N(a) EN(a) zEN(a)nn

[Hint: If Q is bounded 5.3.36 applies. If Q is not bounded and sup I /(z) 1 > M, for some positive E, each m in N, and some Zm in

n Q,

I Zm l > m and II (zm ) 1 > M + E.]

5.6.7 Exercise. (Minimum Modulus Theorem for Coo ) If I E H (Q) and o tJ- I(Q) , for a in Q, I / (a) 1 2': inf I / (z) l . Equality obtains iff I is a constant

zEn function.

[Hint: The function g �f -7 is in H (Q ) ; 5.3.36 applies.]

The set Cn+1 contains the set sn+l of all n + I-tuples (Zl , . . . , zn+d n+l such that L I Zk l 2 > o. There is a relation rv among the elements of sn+l :

k= l {(Z l , . . . , zn+d rv (WI , . . . , wn+d}

{:} {3>.3 /-t{ { 1 >' 1 + I IL I > O} 1\ {>'Zk + ILWk = 0, 1 :::; k :::; n + I } } } .

5.6.8 Exercise. The relation rv is an equivalence.


5.6.9 DEFINITION. pn (C) �f Sn+l / "' . Of particular interest for the context Coo and the map e defined earlier

is p i (C) , the complex projective line.

5.6.10 Exercise. The map

otherwise

is a bijection between p i (C) and Coo .

5.6.11 Exercise. If a E P(f) , for some N(a) and some N(oo),

f [N(a) \ {a}] ::J [N(oo) \ {oo}] ,

v . 5.3.39.

5.6.12 Exercise. If f E H (1[)) , then 0 is a removable singularity of f iff 1R(f) or CS(f) is bounded near o.

[Hint: 5.4.3 and 5.6. 11 apply.]

5 . 7. Contour Integration

The Residue Theorem finds application not only in the theory of C-valued functions defined in some n, but also in the evaluation of definite integrals and in the summation of certain series.

5.7. 1 Example. The integral r 1 2k dx, k E N, is, for positive R, re-JIR 1 + x

lated to the curve (contour)

{ Re27rit /'(t) = 1

-R + (t - 2) 4R . 1 If 0 < t < -- - 2 1 ' if - < t < 1 2 - -

for which /'* is the union of a semicircle r R and an interval [-R, R] . Di

rect calculation shows that if R > 1 , on r R, 1 1 + z2k I 2': R2k - 1 . Hence,

if R is large, I r 1 2k dz l :::; R2:R , whence lim r 1

2k dz = O. JrR l + z - 1 R-too JrR l + z

Thus, if L is the sum of the residues of 1 2k in n+ �f { z : "-5(z) > O } , l + z 1 1 1 then 2k dx = 27riL The set P+ of poles of 2k in n+ is

IR I + x l + z

Section 5.7. Contour Integration 255

1 { z : 'S( z) > 0, 1 + z2k = 0 } . The residues of at the points of 1 + z2k P+ can be calculated via the formulre in 5.4.24.

5.7.2 Exercise. (Jordan's inequality) If 0 :::; t :::; � , then � t :::; sin t .

[Hint: The geometry of the situation provides the clearest basis for the argument .]

sln x 100 •

5.7.3 Example. The integral -- dx can be treated by an integration o x

eiz of - over the curve "( defined in terms of the positive parameters E (small) z and R (large) :

E + 4 (R - E)t

Re47ri(t- i )

-R + 4( - E + R) (t - �) Eei( ( � -t )47r+7r)

.f 1 1 0 < t < - - 4

1 1 if - < t < -4 - - 2 1 3 if - < t < -2 - - 4

. 3 If - < t < 1 4 - -

Thus "(* is the union of two semicircles, Cf and CR in n+ and two intervals

[E, R] and [-R, -E] . Since etZ is holomorphic in the bounded component of z C \ "(*

1 eiz - dz = o. 'Y Z

(5. 7.4)

Rewritten in terms of the four constituent integrals of the left member of (5 .7 .4) , the equation becomes

R . . 1 sin x 1 e"Z 1 etz 2i -- dx = - - dz - - dz.

f X C, Z CR Z

Jordan's inequality, sin x = sin( 7r - x) , and Euler's formula permit the con-

11 eiz z I 1 eiz 10 . i. clusion lim --dz = O. Since - dz = i eue dO, as E -+ 0,

R--+oo CR C, Z 7r

100 . sin x 7r Hence -- dx = _ . o x 2

1 eiz - dz -+ -7ri . c, z


5.7.5 Example. For N in N, if

then "I� is the union of the four complex intervals

1 if 0 < t < -- - 4 1 1 if - < t < -4 - 2 1 3 ' if - < t < -2 - 4

. 3 If - < t < 1 4 -

[ (N + �) (1 - i ) , (N + �) (1 + i )] , [ (N + �) (1 + i ) , (N + �) (-1 + i )] , [ (N + �) (-1 + i ) , (N + �) (-1 - i)] , [ (N + �) (-1 - i ) , (N + �) (1 - i)] .

Direct calculation shows that for the sets

5 �f { 1 - Z

5 clef { resp . 2 = Z

clef { resp. 53 = Z

there are N-free constants CI , C2 , C3 such that

on "I� n 51 I cot 7rZI :::; C1 , on "I� n 52 I cot 7rz l :::; C2 , on "I� n 53 I cot 7rZ I :::; C3 .

Hence on "I� , I cot 7rZ I :::; max {CI , C2 , C3} �f C. Further direct calculation shows that if J E M (C) and P(f) n Z = 0,

then Res n [7r cot 7rZ . J(z) ] = J (n ) , n E Z. If # [P(f)] is finite and

5 �f sum of the residues of 7r cot 7rZ . J(z) at the poles of J,

Section 5 .8 . Exterior Calculus

N for large N, homotopy � J 7r cot 7rZ . I(z) dz = L I(n) + s. 27rl "IN n=- N

257

If 1 E M(C) and for N-free constants K in (0, (0 ) and k in ( 1 , (0) , on "(�, I /(z) 1 :::;

I:k ' then l iN 7r cot 7rZ . I(z) dz l :::; 7r�� (8N + 4) , it follows

upon passage to the limit as N -+ 00 that 00 L I(n) = -so (5 .7 .6)

n = - (X)

If #[P(f)] = No, a second passage to the limit as #(P(f) -+ 00 validates (5.7.6) in general.

5.8 . Exterior Calculus

In Section 5.5, a curve "( ; [0, 1] H Q serves to define a linear functional

;Y ; H(Q) '3 1 H i 1 dz ;

� "( rv "(. Since Q may be viewed as a subset of ffi.2 , 1 �f u + iv is a Coo map

Q '3 (x, y) H (u, v) E ffi.2 . More generally, the curve

"( ; [0, 1] '3 t H [x(t) , y(t)] E Q

defines on the vector space Coo (Q, ffi.2 ) the linear functional ), ( 1 ) according to the formula

),( 1 ) ; Coo (Q, ffi.2 ) '3 [u(x, y) , v(x, y)]

H 11 {u[x(t) , y(t)]x' (t) + v [x(t) , y(t)] y' (t)} dt (5 .8 .1 )

�f i u dx + v dy. (5.8.2)

The notational definition of the right member of (5 .8 .1) as the right member of (5.8.2) is the origin of a formalism that is extended in a manner described below.

When (a, b) E Q, the evaluation map

(a, b) ; Coo (Q, ffi.) '3 k H k(a, b) E ffi.

is a linear functional, an element of [COO (Q, ffi.) ] * . The linear span of the set of all evaluation maps is a subspace VO of [COO (Q, ffi.) ]* . The elements


A(O) of VO are considered to be O-dimensional functionals since they are determined by finite-hence O-dimensional-subsets of Q.

The linear span of the set of all functionals ),(1 ) described in (5.8 . 1 ) i s a subspace Vi of Coo (Q, ffi?) * . The elements A( l ) of Vi are considered to be I-dimensional functionals since they are determined by finite sets of differentiable curves, i .e . , I-dimensional subsets of Q.

When E E S)'2 (Q) and ),2 (E) < 00, according to the formula

),(2) : C(Q, JR.) '3 h H Ie h(x, y) dx dy

E defines a functional ),(2) on C(Q, JR.) . The linear span of the set of all ),(2) is a subspace V2 of [C(Q, JR.)] * . The elements A(2) of V2 are considered to be 2-dimensional functionals since they are determined by finite sets of essentially 2-dimensional subsets of S),(Q) .

An element of f resp. (u , v) resp. h of C(Q, JR.) resp. C (Q, JR.2 ) resp. C(Q, JR.) determines an element w(O) resp. w(l ) resp. w(2) of (VO) * resp. (V I ) * resp. (V2) * according to the formulre

f rv w(O) (A(O) ) �f A(O) (f) ,

resp. (u, v) rv w( l ) (A( I ) ) �f A( l ) [(u, v)] ,

resp . h rv w(2) (A(2) ) �f A(2) (h ) .

In short, the pairings

serve to define two sequences

[C(Q, JR.)] * , [C (Q, JR.2 ) r , [C(Q, JR.)] * , (V0) * , (Vl ) * , (V2) * ,

according as elements in the first resp. second half of a pairing are regarded as functionals defined on the second resp . first half.

Functionals w(o resp . w( l ) resp. w(2) in (VO) * resp. (VI ) * resp. (V2) * are called O-forms resp. I-forms resp. 2-forms. Thus a function f, according to its application, corresponds to a O-form or a 2-form while a pair (u, v) of functions corresponds to a I-form .

• The result of applying a O-form w(O) rv f to the functional determined by (a, b) is f ( a, b) .

Section 5 .8 . Exterior Calculus 259

• The result of applying a I-form w( i ) rv (u, v) to a functional determined by "I : [0, 1] '3 t H [x(t) , y(t)] is

1 {u[x(t) , y(t)]x' (t) + v [x(t) , y(t)]y' (t) } dt

clef 1 d = 'Y U X + v dy . (5.8.3)

• The result of applying a 2-form w(2) rv h to a functional determined by an E in S), is

Ie h(x, y) dx dy �f Ie h dx /\ dy. (5.8.4)

The expression u dx + v dy in the right member of (5.8.3) is the notation for {u[x(t ) , y(t)]x'(t) + v[x(t) , y(t)] y' (t) } dt in the left member. Thus the I-form (u, v) itself is denoted u dx + v dy.

Similarly, the expression h dx /\ dy in the right member of (5.8.4) is the notation for h(x, y) dx dy in the left member. The 2-form h itself is denoted h dx /\ dy.

The virtue of the notations just given becomes clear after the introduction of the operator d defined on O-forms f, I-forms (u, v ) , and 2-forms f according to the formulre:

d : (VO) * '3 w(O) rv f H w( l ) rv (fx , jy ) E (VI ) * , d : (VI ) * '3 w( l ) rv (u, v) H w(2) rv Vx - Uy E (V2) * ,

d : (V2) * '3 w(2) rv f H 0,

for basic forms. The formulre are extended by linearity to all (VO) * resp. (VI ) * resp. (v2) * .

If f(x, y) == X, then df = (1 , 0) ; if g(x, y) == y, then dg = (0, 1 ) :

dx = ( 1 , 0) , dy = (0, 1 ) .

Thus df = 1 dx + O dy = dx, dg = O dx + 1 dy = dy, i .e. ,

dx = dx and dy = dy :

the notation for I-forms is consistent . When ¢ and '1jJ are COO functions on which is based a change of variables

T : ( � ) H [ ¢( �, 1]) ] �f ( x ) 1] '1jJ(�, 1]) y

the conclusions in 4.7. 18 and 4.7.23 lead to the following formulre.


• For a O-form I, I( x, y) r-+ 1 [4>( �, 1]) , 'Ij!( � , 1]) ] �f F( �, 1]) (direct substitution with no modification of the result ) .

• For a I-form (u, v ) , the map T is assumed to be bijectively Coo , i .e. , T- 1 is assumed to be a Coo map. Consequently, if ( � ) �f [X(X, y) ]

1] Y(x, y) ,

for each t, there are unique �(t) and 1] (t) such that

T [ ( � (t) ) ] = [ X(t) ] 1] (t) y(t) ,

i .e. , if J(t) � [� (t) , 1](t) ] , then T maps "I to J. Furthermore,

Hence

u(x, y) = u [4>(� , 1]) , 'Ij!(�, 1]) ] �f U(�, 1])Y,

v (x, y) = v [4>( � , 1]) , 'Ij! (� , 1]) ] �f V (�, 1]) .

u[x(t ) , y(t) ]x' (t ) + v[x(t ) , y(t)]y' (t)

�f [U[�(t) , 1] (t) ]x� + V[�(t) , 1] (t) ]y� ] ((t) + [U[�(t) , 1] (t) ]x1J + V[�(t ) , 1](t) ]Y1J] 1]' (t ) , 1 {u[x(t ) , y(t)]x'(t) + v[x(t) , y(t) ]y' (t) } dt

= 1 {U[�(t ) , 1](t) ]x� + V[�(t) , 1](t) ]Yd ((t) dt

+ 1 {U[�(t ) , 1](t) ]X1J + V[�(t) , 1](t) ]Y1J } 1]'(t) dt,

1 u dx + v dy = 1 [Ux� + Vy�] � + [UX1J + VY1J] d1] :

The I-form u dx + v dy in the variables {x, y} is replaced by the I-form [U x� + Vy�] � + [U x1J + VY1J] d1] in the variables {�, 1]} . In the language of vectors and matrices, [ U(X, y) ] r-+ 8(x, y) . [U(�, 1]) ] � 8(4), 'Ij!) . [U(�, 1]) ]

v ( x, y 8( � , 1] ) V ( � , 1]) 8( � , 1] ) V ( � , 1])

�f J . [U(�, 1]) ] V(�, 1])

(the Jacobian matrix J modifies the result of substitution)

Section 5.S. Exterior Calculus 261

• for a 2-form h, if H(�, 1]) �f h[<p(� , 1] ) , 'Ij! (�, 1] ) ] , then

h(x, y) r-+ det (J) . H(�, 1]) ,

i .e. , h dx 1\ dy r-+ det (J) . H d� 1\ d1] (the determinant of J modifies the result of substitution) .

When z �f x + iy, then dz �f dx + i dy, a complex I-form. By con-. dZ clef d . d H 'f clef 1 (I . ) d b clef 1 (I . ) b ventlOn, Z = x - z y. ence, 1 a = "2 - zg an = "2 + zg , y

polarization, I dx + 9 dy = a dz + b dZ. 5.8.5 Exercise. Both clef 1 . ) d clef 1 (I . I ) Iz dz = "2 (Ix - Zly z and fz dZ = 2 x + z y dZ

are I-forms and dl = Iz dz + fz dZ. The I-form conjugate to w( 1) �f I dx + 9 dy is *w( 1 ) �f -g dx + I dy.

The form conjugate to dP = Px dx + Py dy is *dP = -Py dx + Px dy. When "( : t r-+ "((t) E C is a rectifiable curve and w( 1 ) �f P dx + Q dy,

then 1 W( l ) �f 1 1 {Pb(t)] �: + Qb(t) ] ��} dt = 1 P dx + Q dy.

Forms (also known as differential forms) can be added, multiplied, differentiated, and integrated according to the following procedures .

• Addition. The sum of: • two O-forms or two 2-forms I and 9 is the O-form or 2-form

I + g;

• two I-forms I dx + 9 dy and p dx + q dy is the I-form

(I + p) dx + (g + q) dy;

• Multiplication. The product of: • two O-forms I and 9 is the O-form I · g; • a O-form k and a I-form I dx + 9 dy is the I-form

kl dx + kg dy;

• a O-form k and a 2-form I is the 2-form kl; ( 1 ) clef ( 1 ) clef d .

• two I-forms WI = I dx + 9 dy and w2 = p X + q dy IS the 2-form wil l 1\ W�l ) �f (lq - pg) dx 1\ dy, the exterior (wedge) product of wi 1) and W�l ) ) .

• Differentials. The differential of: • a O-form I is the I-form Ix dx + Iy dy;


• a I-form f dx + g dy is the 2-form (gx - fy ) dx 1\ dy; • the product kw( l ) is the 2-form k · dw( l ) + dk · w( l ) .

• Integration. The integration of:

• a O-form f over a complex interval [a, b] is 11 f((1 - t)a + tb) dt;

• a I-form f dx + g dy over a curve "I : [0, 1] 3 t r-+ [a(t) , ,6(t)] E ffi.2 IS

11 fb(t)] dx h') + gb(t)] dy b'(t)]

= 11 fb(t)]a'(t) dt + gb(t)],6'(t) dt,

i.e. , the line integral 1 f dx + g dy;

• a 2-form w (2) over E is Ie w (2) dx 1\ dy.

[ 5.8.6 Note. The preceding discussion deals only with two variables and 0- , 1-, and 2-forms. The structure may be developed for n variables and n-forms, n E N. For example, when n > 2, the product of a I-form and a 2-form and the differential of a 2-form are 3-forms, etc. In a natural way, when n variables are involved and 0 < k E N, all (n + k )-forms are zero. Excellent references are [Hic, Lan, SiT, Spi] .]

5.8.7 Example. Green's Theorem (Stokes 's Theorem for the plane) may be stated in terms of forms as follows.

If R �f [a, b] X [c, d] c fl, then

r + r + r + r f dx + g dy, i[(a,c) , (b,c) ] i[(b,c) , (b,d) ] i[(b,d) , (a,d) ] i[(a,d) , (a,c) ]

= r (gx - fy ) dx 1\ dy, (5.8.8) i[a,b] x [c,d]

r w( l) = r dw(l) . (5.8.9) ia(R) iR

A proof of Green's Theorem for well-behaved functions defined over rectangles in ffi.2 is given by direct appeal to the FTC. Indeed, Fubini's theorem implies

r (gx - fy) dx 1\ dy = rb (Jd

(gx - fy ) dY) dx. i[a,b] x [c,d] ia c

(5.8.10)

When FTC is applied to the right member of (5.8. 10) , the result is (5.8.9) .

Section 5.S. Exterior Calculus 263

The equation (5.8.9) itself can be related to FTC, e.g. ,

h(q) - h(p) = lq h'(x) d)",

according to the following interpretation. In JR., the boundary a ([P, q] ) of the interval [p, q] is {p, q} . For the

signed measure space (a([p, q] ) , !fj({p, q}) , /-l) such that { I if y = q /-l : a([p, q] ) 3 y r-+ l ' f ' - 1 Y = p

if h' exists at every point of [p, q] while h' E Ll ([P, q] , ).. ) , the burden of FTC is r h d/-l = 1 h' d).. . J a[p,q] [p,q]

Stripped of all its qualifiers, Stokes's Theorem says that the result of integrating a function h over the boundary as of a set S is the same as the result of integrating some kind of derivative of h over S itself.

5.8. 11 Exercise. For I-forms wP ) , W�l) : a) wi l l 1\ W�l ) = _W�l ) 1\ wP ) ; b) ( 1 ) 1\ ( 1 ) - 0 WI WI -

• [Hint: If T ( ; ) = ( � ) , then det (J) = - 1.]

[ 5.8.12 Note. Hence, by definition,

Iv F(x, y) dx 1\ dy = - Iv F(x, y) dy 1\ dx.]

5.8. 13 Exercise. If u E COO (V, q, then

r u dz = 2i r au dx 1\ dy = r au dZ 1\ dz. Jav Jv Jv

5.8. 14 THEOREM. (Pompeiu) IF V IS A NONEMPTY OPEN SET SUCH THAT FOR CONTINUOUSLY DIFFERENTIABLE JORDAN CURVES {'Yih:'O:i:'O:IEN '

I av = U 1': i= l

AND u HAS CONTINUOUS FIRST DERIVATIVES ON Ve , FOR W IN V, 1 {fa u (z) 1 au } u( w ) = -. -- dz + -- dz 1\ dZ . 27rl av z - W v z - W (5.8.15)


[ 5.8.16 Remark. If u E H (V) , then au = 0 and (5.8. 15) is Cauchy's integral formula (5.4 .14) . Thus the second term in the right member of (5 .8 .15) may be regarded as an error term used, when u tt H (V) , to compensate for the failure of u to be holomorphic and for the concomitant potential failure of Cauchy's formula.]

PROOF. If E < d[w, (C \ V)] and U, �f { z

1 z r-+ -z - w

z E V, I z - w i > E } , then

is holomorphic in Uf o Stokes 's Theorem applied to av 3 z r-+ u(z) yields z - w

au

J Oz az A dz = r u(z) dz - l27r U (W + EeiO) i dO.

u , z - w i av z - w 0

As E + 0 the formula emerges.

[ 5.8.17 Note. The treatment above of Pompeiu's formula appeals to Stokes 's Theorem, which is derivable (v. [Lan] , [Spi] ) in the current context by methods related those used to reach, e.g. , 5.3 .11 . Although Stokes 's Theorem implies 5.3 .11 and others (v. [Ho] ) , the theory of differential forms [Lan, SiT, Spi] lies at the heart of the matter.

On the other hand, in the current treatment , appeal is made to Pompeiu's formula in only the simplest instances , e.g. , 7. 1 .16, where the validity of Stokes's Theorem is directly demonstrable, v. 5.9. 15.]

o

5.8.18 Exercise. a) du *dv = dv *du. b) l1 = d (*d) . c) If QI C Q, QI is relatively compact in Q, and aQI is a rectifiable curve, e.g. , if r > 0 and QI = D(a, rt C Q, there emerges Green's formula

r u *dv - v *du = r (u l1v - v l1v) dx A dy. ian, in,

[Hint: Stokes 's Theorem applies .]



5.9. 1 Exercise. If {fn}nEN C H (1U) , J E C (1U, C) , and for each r in (0, 1 ) ,

then J E H (1U) . 12". lim IJn (reiO) - J (reiO) I dO = 0, n-teXJ 0

[Hint: Cauchy's integral formula implies that if z E 1U, then

lim In(z) = J(z ) . n---+=

Vitali's Theorem 5.3.60 applies .]

5.9.2 Exercise. If Q �1Uc , a E '][', J E H (Q \ {a}) , a E P(f) , Ord f (a) = 1 , =

and in 1U, J(z) = L cnzn, then lim � = a. n=O n---+= Cn+ l

[Hint: The Cauchy-Hadamard Theorem applies to (z - a)J(z) .]

5.9.3 Exercise. If J E H [D(O, 2r] and the order of each zero of J in D(0, 2r is one, what are the zeros of J in 1U if

r I'(z) _ r I' (z) _ r 2 J'(z) _ .? J.lf J(z) dz

- J.lf Z J(z) dz - J.lf

Z J(z) dz - 7n .

5.9.4 Exercise. If J E H (Q) and a E Q, the radius oj convergence ra of CXJ JCnl (a) the power series '" (z - a)n is at least inf { l a - w i : w E aQ } . � n! n=O

5.9.5 Exercise. Every region Q contains a maximal simply connected subregion, i.e. , a simply connected subregion Q1 that is properly contained in no simply connected subregion of Q.

[Hint: If P is the nonempty set of simply connected subregions of Q, P is a poset with respect to {A -< B} {:} {A C B}. Zorn's Lemma applies .]

5.9.6 Exercise. For some region Q, there is no unique maximal simply connected subregion of Q.

[Hint: The region Q �f 1U \ {O} merits attention.]

5.9.7 Exercise. If K is compact , U is open, and K c U e Q, there are constants {Crn}rnEZ+ such that for each J in H (Q) ,


[Hint: If '1jJ is infinitely differentiable, vanishes off a compact subset of U, and '1jJ == 1 in an open set V containing K, then

Pompeiu's formula yields

'1jJ(w)f(w) = � J f(z)8'1jJ(z) dz 1\ az. 27r� u z - w (5 .9 .8)

Since 8'1jJ = 0 in V, if z is such that 8'1jJ -j. 0, for some positive E,

I z - wi 2': E whenever w E K. Differentiation of (5.9.8) and 5.8.13 yield the result .]

5.9.9 Exercise. For the operators 8 and 8 and a differentiable function f �f P + iQ, there obtain the following formulre.

8f 8f -8x dx + 8y dy = 8f dz + 8f az,

8U 0 g) = [(81) 0 g] . 8g + [ (8f) ] · 8g, 8U 0 g) = [(81) 0 g] . 8g + [ (8f) ] . ag.

[Hint: If f(z) �f f(x, y) and g(z) �f p(x, y) + iq(x , y) , then

f 0 g(z) = f[P(x, y) , q(x, y) ] .]

5.9.10 Exercise. a) If r > 0 and f E {H [D (a , rr ] n C (D(a, r) , C] }, there is a sequence {Pn} nEN of polynomial functions such that on D( a, r) , Pn � f.

b) If Q �f D(O, lr \ {O} and f(z) �f �, then f E H (Q) but no sequence z of polynomial functions converges to f uniformly on compact subsets of Q.

[Hint: a) Fejer's Theorem (3.7.7e) ) and the Maximum Modulus Theorem (5.3.36) apply.]

5.9. 11 Exercise. In 5.3.43 a satisfactory value for p is :

5.9.12 Exercise. A net of functions is locally uniformly convergent resp. locally uniformly Cauchy resp. locally uniformly bounded on an open set U iff U is the union of open sets on each of which the net is uniformly convergent resp. uniformly Cauchy resp. uniformly bounded.


5.9. 13 Exercise. (Montel) A subset F of H(Q) is normal iff F is locally bounded, i.e. , iff for each compact subset K of Q and some MK in JR.,


5.9. 14 Exercise. If: a) l'(t) = e27rit , O :S; t :s; 1 ;

sup I f(x) 1 :s; MK. x E K J E F

b) for some ¢ in C1 (D(0, 1 ) , q , ¢[D(O, 1 )] �f QC and for l' as in a) ,

and ¢ 0 l' is a Jordan curve such that aQ = (¢ 0 1')* ; c) f and g in Q 3 (x, y) r-+ [J(x, y) , g(x, y)] E JR.2 are continuously differ

entiable Stokes 's Theorem in the form (5.8.6) is valid.

5.9. 15 Exercise. If {u, v} c COO(1U, JR.) , det [��::�n > 0, and

clef Q = { [u(x, y) , v(x, y)] : -0.5 < x, y < 0.5 } ,

then Q is relatively compact and for any I-form w( l ) , r W(l ) = r dw( l ) . lo(n) ln [Hint: Both 5.9. 15 and the discussion in 5.8.5 apply.] [ 5.9.16 Remark. The result 5.9.16 provides extended circumstances where Stokes 's Theorem applies . An argument based on patching together squares like Q �f { (x, y) : -0.5 < x, y < 0.5 } and their images under maps like

T : Q 3 (x, y) r-+ [u(x, y) , v(x, y)] E Q C JR.2

above leads to very general forms of Stokes's Theorem [Lan, Spi] .

If ),2 (Q) = 00, In dw(l) need not make sense. For example, if

Q �f { z : I z l > I } , then ),2 (Q) = 00 (and Q is not relatively compact) . For the I-form w( l ) �f -y dx + x dy, dw( l ) = 2 dx 1\ dy and

r w( l ) = 27r -j. r dw(l ) = 00.] lo(n) ln


5.9. 17 Exercise. a) There is a function I in H (D(O, It ) and such that I (�) = I ( -�) = :2 ' b) There is no function g in H (D(O, 1 )0 ) and

such that g (�) = g ( -�) = :3 ' 5.9. 18 Exercise. If I E H (fl) and 5 �f { z : I I(z) l :S; I } c fl, either I is a constant or Z(f) n 5° -j. 0.

5.9. 19 Exercise. If I is entire, then: a) 1 I�Z) b dz exists for

I z l=r (z - a (z - ) all large r. b) If I I I is also bounded, then for each a, J'(a) = ° (a second proof of Liouville's Theorem.)

00 5.9.20 Exercise. If L anzn converges in 1U and: a) ao -j. 0, what is

n=Q 00

the recursion formula for the coefficients in L bnzn that represents the n=O

(reciprocal) function g such that near 0, g . I(z) == 17 b) al -j. 0, what is 00

the recursion formula for the coefficients in L cnzn that represents the n=O

(inverse) function h such that near 0, h 0 I(z) == z7

5.9.21 Exercise. If I(x, y) �f u(x, y) + iv(x, y) and both u and v are d· U" • bl clef . f . R luerentm e at z = x + zy, or some nonnegatIve , { I(z + h� - I(z) : h -j. ° r = 8[D(a , R)] .

clef °0 clef o</> [Hint: For z = re' and h = pe' , p > 0, ¢ fixed, the calculation I(z + h) - I(z) of h as p + ° applies .]

5.9.22 Exercise. HI (10, z) � Z.

[Hint: For k in Z and "Ik : [0 , 1] 3 t r-+ (0.S)e2k".it , "Ik = k::;i . If 1] is a rectifiable loop and 1]* C 10 then for some k in Z, Ind 1J (O) = k. If I E H (10) , then I is representable by a Laurent series .]

5.9.23 Exercise. a) If

"Il (t) �f -1 + e27ri ( l -t) , "I2 (t) �f 1 + e27rit , fl �f C \ ( {-I} U { I} ) ,

and r �f ::;i - ,.y;, then r � n O. b) For some curves 1]1 and 1]2 , s: clef - 1 d s: clef - 1 U 1 = 1]1 . "11 . 1]1 an u2 = 1]2 . "12 . 1]2


are loops and 81 (0) = 82 (0) . c ) {<t>} �f {8d {J2 } {8� 1 } {82 1 } -j. 1 but h ({<t>}) = 0 (in HdQ, Z)] , v. 5.5.28.

5.9.24 Exercise. The conclusion in 5.3.62 can fail if the condition on Q is replaced by on a set S such that S e Q and Q ::) S· -j. (/) as in 5.3.52.

. zn + 1 clef 5.9.25 Exercise. If I z l -j. 1 and hm -- = G(z) , then G E H (C \ 1I') , n---+= zn - 1

and { -I G(z) = 1 if I z l < 1 if I z l > 1 .

ex:> 1 5.9.26 Exercise. If �(z) > 1, then the Dirichlet series '" - converges � nz n=1 and defines a function f holomorphic in Q �f { z : �(z) > 1 }.

[ 5.9.27 Remark. Riemann's zeta function ( is defined and holomorphic in C \ {I} and ( I n= f. Furthermore, P(() = { I } and Ord « (l ) = 1. It is known that Z(() C { a + it : O .:s: a .:s: I } and that Z (() is symmetric with respect to both { a + it : a = � } and JR.. Riemann conjectured that Z (() C { a + it : a = � } . As of this writing, his conjecture remains unresolved, despite the efforts of some of the greatest analysts since Riemann's time. Riemann's zeta function is of central importance in number theory, particularly in the study of the distribution of prime numbers, i.e. , the cardinality 7r( x) of the set of prime natural numbers n such that n .:s: x. In 1896, J. Hadamard and C.-J. de la Vallee Poussin, using properties of (, independently proved

. 7r(x) THE PRIME NUMBER THEOREM. hm -( x ) = 1 . x---+=

ln x

In 1948, P. Erdos and A. Selberg [Sha] proved the Prime Number Theorem without recourse to the methods of complex analysis.]

6 Harmonic Functions

6.1. Basic Properties

The subject of harmonic functions appears in 5.3.57. The conclusion to be drawn from 5.3.59 is that for some regions Q and some u in HalR (Q) , there is in HalR(Q) no function v that serves as a harmonic conjugate to u throughout the region Q. The next results explore other possibilities.

6.1 .1 Exercise. a) If I �f u + iv E H (Q) , then

I E Ha(Q) and {u, v} C HalR (Q).

b) There are in HalR (Q) functions u and v such that u + iv tt H (Q) .

6.1.2 LEMMA. a) IF u E Ha (Q) AND D(a, rr C Q, THEN THERE IS A v SUCH THAT I � u + iv E H [D(a, rr] ' i.e. , CONFINED TO D(a, rr , v IS A HARMONIC CONJUGATE OF u. b) IF, CONFINED TO D(a, rr, VI AND v2 ARE HARMONIC CONJUGATES OF u, THEN FOR SOME REAL CONSTANT C, VI - v2 = C. PROOF. a) If u E HaIR (D(a, rr) and a �f a + i(3, direct calculation, in view of the existence of Uxx and Uyy and the validity of t:m = 0, shows that the function v : D(a, rr 3 x + iy r-+ iY ux (x, t) dt - I

x Uy (s, (3) ds is such

that T : D(a, rr 3 (x, y) r-+ [u(x, y) , v(x, y)l has a derivative and furthermore that Ux = Vy and uy = -vx . Thus I � u + iv E H [D(a, rrl .

If u = �(u) + i<S(u) �f p + iq E Ha [D(a, rt] ' then

{p, q} c HaIR [D(a, rn .

The previous argument implies that for some � and 1] in HaIR (D(a, rr) , {p + i� , q + i1]} C H [D(a, rt] ' whence

p + iq + (i� - 1]) �f u + iv E H [D(a, rtl .

b) If Ij �f u + iVj E H [D(a, rtl , j = 1 , 2, then i (II - h) is JR.-valued in (D(a, rt) and the Cauchy-Riemann equations imply that for some real constant c, i (II - h) = -c, i.e. , VI - V2 = C. 0

270

Section 6 .1 . Basic Properties 271

6.1.3 COROLLARY. IF u E HalR (Q) , a �f a + i(3, AND D (a , rr c Q: a) THERE IS A SEQUENCE {PTnn} :,n=o OF CONSTANTS SUCH THAT IN D( a, r r ,

00 , 00 u(x, y) = L PTnn (X - a) Tn (y - (3)n ; b) THERE ARE SEQUENCES {Cn }nEZ '

Tn ,n=O {an}nEZ+ ' AND {bn}nEZ+ SUCH THAT IF 0 :::; R < r, THEN

00 00 u (a + ReiO) = L cnRneinO = L anRn cos nO + bnRn sin nO.

n==-(X) n=O

PROOF. The argument in the Hint following 5.3.25 applies. 6.1.4 Exercise. If r > 0 and u E HaIR [D (a , rr ] ' for some v ,

v + iu E H [D (a , rr ] .

6.1 .5 LEMMA. IF u E Ha (Q) , D (a , rr c Q, AND 0 :::; R < r , THEN

o

1 12". u(a) = - u (a + ReiO) dO, 27r 0 (6 .1 .6)

i.e. , u ENJOYS THE Mean Value Property MVP AT EACH POINT OF Q. [ 6.1. 7 Remark. Customarily the symbol MVP(Q) is reserved for the set of functions continuous in Q and enjoying the Mean Value Property at each point of Q; there is a corresponding meaning for MVPIR(Q). Thus 6.1.5 may be viewed as the assertion: Ha (Q) c MVP(Q) . The reversed inclusion is the burden of 6.2.16 below.]

PROOF. For the harmonic conjugate v that serves in D (a , rr ,

f �f u + iv E H [D(a, rr] .

Cauchy's formula applies.

6.1.8 THEOREM. (Maximum Principle) IF u E MVPIR (Q) , a E Q, AND

o

u(a) 2': sup { u (x, y) : (x, y) E Q } , (6. 1 .9) THEN u IS A CONSTANT FUNCTION. PROOF. The MVP asserts that if D(a, R) C Q, the value of u at the center of D( a, R) is the average of its values on 8[D( a, R)] . Hence, if mR resp. MR are the minimum resp. maximum of u on 8[D( a, R)] , then (6. 1 .6) implies

272 Chapter 6. Harmonic Functions

mR :::; u(a) :::; MR and (6. 1 .9) implies that for all R as defined, u(a) = MR. 2 2 clef 2 2 ) If Xo + Yo = s :::; R and u (xo , Yo ) < u(a , then

{ (x, y) : X2 + y2 = S2 , U(X, y) < u(a) }

is a nonempty open subset of 8[D(a, s) ] . Thus

1 12". u(a) = - u (a + seiO) de < u(a) , 27r 0 a contradiction. Hence, if u is not constant , for any a in fl, u(a) cannot be a local maximum value of u: (6. 1 .9) is denied. 0 6.1 .10 Exercise. If u E HaIR [D( a, r r] n C[D(O, r) , C] , then (6 .1 .6) is valid when R = r .

[Hint: When R < r , 6.1 .5 applies . Passage to the limit as R t r is justified by the Dominated Convergence Theorem (2 .1 . 15) and 6.1 .8.]

6 .1 .11 THEOREM. (Maximum Principle in Coo ) IF fl c C, u E MVPIR(fl) , AND

{ (a, b) E 8oo (fl) } '* { inf sup u(x, y) :::; o} , N[(a,b) ] EN[(a,b)] (x,y) EN[(a,b) ]nn

THEN u = 0 OR u(fl) c (-00 , 0) .

PROOF. If the result is false there are two possibilities : a) for some (xo , Yo ) in fl, u (xo , Yo ) > 0; b) for some (xo , Yo ) in fl, u (xo , Yo ) = 0 while u(fl) C (-00, 0] .

If a) is true, for some positive E,

If K is unbounded, then 00 E 8oo (fl) and for a sequence { (xn , Yn ) }nEN in K, (xn , Yn ) -+ 00 as n -+ 00. Hence lim u(x, y) 2': E > 0, a contradiction of

(x ,y)---+oo the hypothesis. Hence K is bounded and since u is continuous , K is closed: K is compact .

Thus, for some (p, q) in K, u(p, q) = max { u(x, y) : (x, y) E K } 2': E.

If (x, y) E fl \ K, then u(x, y) < E, whence

u(p, q) = max { u(x, y) : (x, y) E fl } ,

and 6.1 .8 implies u = u(p, q) 2': E, a contradiction of the hypothesis .

Section 6.2. Functions Harmonic in a Disc 273

If b) is true, 6.1 .8 implies u == o. o 6.1.12 Exercise. If {u , v} c MVPIR(fl) and for each point (a, b) in 8oo (fl)

inf sup u(x, y) N[(a,b) ]EN[(a,b) ] (x,y) EN[(a,b) ]nn

� sup inf v(x , y) , N[(a,b) ] EN[(a,b) ] (x,y) EN[(a,b) ]nn

on fl, either u < v or u == v . [Hint: 6.1 .11 applies to u - v .]

6.1 .13 Example. a) Although f : C 3 x + iy r-+ x is in HalR (C) , f2 is not harmonic. a) both g : C 3 z r-+ eZ and g2 are harmonic. c) The map h : C 3 x + iy r-+ y is in HalR (C) ; h2 is not harmonic but f2 - h2 E HalR (C) .

6.1 .14 Exercise. If f E H (fl) and I f I E HalR (fl) , then f is a constant function.

[Hint: Since bolf l = 0, if f �f u + iv , then Ux + Vx = Uy + Vy = o. The Cauchy-Riemann equations imply !, == 0.]

6.1 .15 Exercise. If f E H (fl) and u E Ha [J(fl) ] , then u 0 f E Ha(fl) . [Hint: If a E fl, for some positive r and positive s,

D(f(a) , st C J [D(a, rn c f(fl),

and 6.1.2 implies for some v, g �f u + iv E H { [D(f(a) , st] }, i .e . , g o f E H (fl) .]

6.2. Functions Harmonic in a Disc

1 00 00 The function f : 1U 3 z r-+ � = (1 + z) '" zn = 1 + 2 '" zn is in H (1U) . 1 - z � �

n=O n= l

Hence 'iR(f) E HaIR (1U) . Customarily, polar coordinates are used to discuss 'iR(f) in 1U, and when z = reiO , 0 :::; r < 1 , fJ E JR.,

1 + reiO 1 - r2 2 sin fJ f ( z ) = = + i -....."...---,------;,-1 - reiO 1 - 2r cos fJ + r2 1 - 2r cos fJ + r2 '

'iR { l + reiO } = 1 - r2 �f Pr(fJ) 1 - reiO 1 - 2r cos fJ + r2 .

The map 1U 3 (r, fJ) r-+ Pr (fJ) is the Poisson kernel .


6.2. 1 Exercise. 00

n= - (X)

6.2.2 LEMMA. IF 0 :::; r < R AND Ca (R) �f { z : I z - a l = R } , FOR THE COMPLEX MEASURE SPACE (Ca (R), 5(3 [Ca (R)] , p,) ,

h : D(a, rt 3 reiO r-+ 1 Pfi ( 8 - t ) dp, (a + Reit ) �f h(r, 8) Ca rr)

IS IN Ha [D(a, Rt] .

PROOF. Since PI- (8 - t ) = � { eit + �} , PI- is in HaIR (D(O, Rt) . Be-R . re' R e,t - -R

cause p, = �(p,) + iSS(p,) �f a + i(3, when z = reiO , h is a complex linear combination of the real and imaginary parts of cl f r27r Reit + z . r27r Reit + z .

I(z) � Jo Reit _ z dO' (a + Re't ) + i Jo Reit _ z d(3 (a + Re't ) .

By virtue of 5.3.25, each integral above represents a function holomorphic in D(a, Rt · 0

[ 6.2.3 Remark. On Ca(R) arc-length may be used as a basis for a measure space [Ca (R), 5(3 [Ca (R)] , �] that is the analog of T in 4.5.2. The group '][' acts on Ca (R) according to the rule

'][' x Ca (R) 3 {eiO , z } r-+ a + eiO (z - a) .

With respect to the action of '][' on Ca(R), � i s action-invariant : if E E 5 (3 [Ca (R)] and 0 :::; 8 < 271", then

a + eio (E - a) E 5(3 [Ca (R)] , � [a + eio (E - a)] = �(E) . clef dp, If p, « � and g = � ' by abuse of language, 6.2.2 says the con-

volution Pfi * g is harmonic .

When k defined on Ca(r) is such that

r27r Jo Pfi (8 - t)k (a + reit ) dt

Section 6.2. Functions Harmonic in a Disc

exists , the result is the Poisson transform of k and IS denoted P(k) . It is a harmonic function defined in D (a, rr .

The role of the family {PI.. } < R is discussed in Section 6.4.] R 0_ 1'<

275

6.2.4 Exercise. The following are alternative formulre for P -Ii ((J - t) when z = ReiO and a = reit :

PI.. ((J _ t ) = 1R ( z + a) = R2 - 1 a 1 2

R Z - a I z - al 2 R2 _ r2

R2 - 2rR cos((J - t) + r2 '

6.2.5 THEOREM. IF f E H (1U) n C [(1U)C , C] AND, FOR

clef "0 z = ret , 0 :s; r < 1 , 0 :s; (J < 27r, 00 00

THE REPRESENTATION f(z) = L CnZnOBTAINS THEN:a) L I cn l 2 < 00; b) n=O n=O (POISSON' S FORMULA )

PROOF. a) Since f lTE L2 ('][', T) and {eint } nEZ consists of pairwise orthogonal functions of absolute value one, Bessel's inequality (v. 3 .2 .14c) ) implies

Cauchy's formula for the Cn applies. b) In particular, the argument in a) implies

(6.2.6)

According to Cauchy's formula, if I z l < 1 , then

(6.2 .7)

(6.2.8)


Then (6.2.7) and (6.2.8) imply

_ � 12". 1 (eit ) � 12". 1 (eit ) eitz I (z ) - . dt + . dt, 27r 0 1 - e-,tz 27r 0 1 - e,tz

1 r2". . ( 1 rei(t-O) ) = 27r Jo 1 (e't ) 1 - rei(O-r) + 1 _ reit-O)

= � r2". Pr((} - t )1 (eit) dt . 27r Jo

6.2.9 THEOREM. IF u E HaIR [D(O, Rn n C[D(O, R) , C] AND l a l < R,

u(a) = � r2". R2- l a 1 2

2 U (Rei</» d¢. 27r Jo IRe'</> - al

c clef R(R( + a) clef PROOF. For the map I : (1U) 3 ( r-+ I(() = ( = z, R + a

1(1I') = Ca (R ) , 1 [(1U)C ] = D(O, R), and 1(0) = a.

o

Thus u o l E HaIR (1U) n C [(1U)C , C] (v . 6.1 . 15) and 6.1 .10 implies that if ( = eiO and z = 1 (eiO ) �f Rei</> , then

. R(z - a) . d( Smce ( = 2 and d() = -l- , R - az (

d() = (

z � a + R2 a� az ) d¢ = (R::i� a + R2 a�:;ei</> ) d¢ = R2 - l a l 2 d¢ (= � (�) d¢) . 0 I z - a l2 z - a

6.2.10 Exercise. a) In the context of 6.2.9 with D(O, Rt replaced by . 1 12". .

D(a, Rt , if r < R, u (a + re'O) = - Pf; ((} - ¢)u (a + Re'</» d¢. b) If 27r 0 D( a, r) C fl, 1 E H (fl) , Z(f) n D( a, r) = {a1 , . . . , aN} (a zero of order k is listed k times ) , and I (a) i- 0, then

(6.2 . 1 1 )


[Hint: For some {rn}nEN and all fJ, 0 < rn t r and f (rneiO) -j. O. Poisson's formula applies .]

[ 6.2.12 Remark. The result (6.2 . 1 1 ) is the Poisson-Jensen formula.]

277

In Section 6.4, the properties a )-c) listed below for the Poisson kernel are explored in a general context . Analogies are drawn between Fejer's kernels (3.7.6) and approximate identities, which play an important role in topological algebras .

6.2.13 THEOREM. FOR THE POISSON KERNELS THE FOLLOWING OBTAIN:

a) Pr (fJ) > 0, O :S; r < 1, fJ E JR.; 1 12". b) IF O :S; r < 1 , THEN - Pr (fJ ) dfJ = 1 ; 27r 0

c) IF 0 < fJ < 27r, THEN lim Pr (fJ) = 0 AND THE LIMIT IS UNIFORM IF rtl

o < r5 :s; fJ :s; 27r - r5 .

PROOF. a) Since 1 - 2r cos fJ + r2 = (1 - r)2 + 4r sin2 �, Pr(fJ) is the quotient of two positive functions.

b) The result in 6.2.5b) applies when f (z ) == 1 . I 7r 2 ( • c) If 0 :s; r < 1 and fJl :s; "2 ' then 0 < 1 - r < 2 1 - r) and, owmg to

Jordan's inequality (5.7.2 ) , (1 - r)2 + 4r sin2 � > (1 _ r)2 + 4r�2. Thus 2 7r

7r if "2 < I fJ l < 7r

[ 6.2. 14 Note. If sin � = 0, e.g. , if fJ = 0 or fJ = 27r, Pr (fJ) t oo as r t 1 .]

6.2.15 COROLLARY. IF f E C(1I', q AND

THEN u IS THE UNIQUE FUNCTION SUCH THAT

u E Ha (1U) AND u (r, fJ) � f (eiO) AS r t 1 .

o


PROOF. By virtue of 6.2.2, u E Ha (1U) . Furthermore, 6.2.13b) implies

I I (eiO) - u (r, O) 1 = i 2� 127r {I [eiO] - I [ei(O-t) ] } Pr(t ) dt i

< � ('s

+ � 1

27r-8

- 27r Jo 27r 8 + 2� 1:�8 1 {1 [eiO] - I [ei(O-t) ] } Pr(t) 1 dt

�f I + II + III.

If 0 < r5 :s; t :s; 27r - r5, since I is continuous, 6. 2. 13c) implies, that for positive E and r near and below 1, I I < E. The continuity of I and the non-

negativity of Pr imply that if r5 is small, I + III :S; E� r27r Pr(t ) dt = E. 27r Jo The uniqueness of u is a consequence of the Maximum Principle. 0 The preceding developments lead to the following characterization of

harmonic functions .

6.2.16 THEOREM. MVP(Q) = Ha(Q) . PROOF. According to 6.1 .5, Ha(Q) c MVP(Q).

If I E MVP(Q), D(a, R) C Q, 0 :s; r < R, and

then 6.2.2 implies F E Ha (D(a, Rt ) . Furthermore, 6.2.15 implies

F (a + Reit ) = I (a + Reit ) ,

i.e., 'iR(F - f) and 'J(F - I) are in MVPIR (D( a, Rt) and on 8[D( a, R)] each is zero. The Maximum Principle implies F - I = 0 on D( a, R) . 0

6.2.17 LEMMA. IF

Q c n+, a < b, 8(Q) n ffi. = [a, b] , � -

I Q1 = Q U (a, b) U Q, I E H(Q) n C[Q U (a, b), q , AND CS(f) (a,b) = 0,

IN H (QI ) THERE IS A UNIQUE II SUCH THAT II Inu(a,b) = I· [ 6.2.18 Remark. The result 6.2. 17 i s a version of the Schwarz Reflection Principle . A variation on the theme is explored in Chapter 8, where the geometry of reflections and inversions in


circles is discussed, and in Chapter 10, where the relevance of the Schwarz Reflection Principle to the process of analytic continuation is illuminated.]

279

PROOF. The Identity Theorem (5.3.52) implies there is at most one function h as described. If

{ I(z) h (z) = _

1 ("2)

if z E Q u (a, b) if z E n

then h E C (QI , q n H (Q u n) . If a < x < b, r > 0, D (x , rt c QI , and

for large n, r h (z) dz = 0 and Ja(Sn, ± )

1 h (z ) dz = }�+� ( r + r h (Z) dZ) = O. I z-x l=r Ja(Sn , + ) Ja(Sn, _ )

Morera's Theorem (5.3.48) applies . 6.2.19 Exercise. For Q in 6.2. 17, if x E (a, b) , for some positive r,

D(x, rt n Q = D(x, rt n n+ .

o

(The preceding positive statement is an implicit ingredient of the PROOF above. )

[ 6.2.20 Note. The conditions

8(Q) n lR. = [a, b] and 1 E H (Q) n C[Q u (a, b), C] are intended to preclude the possibility that [a, b] is, for I, part of its natural boundary (v. Hint following 7.1.28) .]

The interplay between the theory of locally holomorphic functions and the theory of harmonic functions emerges in the next result , which is a considerable strengthening of 6.2.17.

6.2.21 THEOREM. FOR Q, a, b, AND QI AS IN 6.2. 17, THE CONCLUSION OF 6.2.17 IS VALID IF 1 E H (Q) , 2s(f) � v E C [Q u (a, b) , lR.] , AND v l (a,b)= O.


PROOF. If x E (a, b), for some positive r, D(x, rt c Ql . If

{ -v (z) if z E Q v( z) � v( z) i f z E Q ,

o if z E (a, b) the integral formula that defines the Mean Value Property implies that v E MVPIR (Ql ) , whence v E HaIR (Q1 ) .

Hence, if a < x < b, for some nonempty D(x, rt there is a u that is a harmonic conjugate of v in D(x, rt: u + iv E H (D(x, rt ) . Moreover, 1 - (u + iv) is JR.-valued in D(x, rt n n+ , whence, for some real constant C, 1 - (u + iv) == C in D(x, rt n n+ . Thus u + c is also a harmonic conju-gate of v. Hence hx �f u + c + iv is holomorphic in D(x, rt and coincides with 1 in D(x, rt n n+ . Since hx is JR.-valued on D(x, rt n JR., if, for z

00 in D(x, rt, hx(z) = L cn (z - x)n , each Cn is real. Hence in D(x, rt ,

n=O 00

hx (z) = L Cn (z - xt = hx (z) . If a < y < b and D(x, rt n D(y, st -j. 0, n=O

then hx = hy in D(x, rt n D(y, st n Q and thus, by virtue of the Iden-tity Theorem (5.3.52 ) , hx = hy in D(x, rt n D(y, st . Direct calculations show that if { I(z) if z E Q

h (z) �f hx(z) if z E D(x, rt , 1 (z) if z E Q

then h is consistently defined and meets the requirements. 0 [ 6.2.22 Note. The continuity of I on Q u (a, b) is not part of the hypothesis of 6.2.21.]

6.2.23 Exercise. a) (Schwarz's formula) If

h E HaIR [D(a, Rt] n C[D(a, R) , JR.] , def l w + z h(w) [ ] . then I(z) = -- -- dw E H D(O, Rt . b) (Harnack's mequa-Iw l =R w - z w

lities ) If h E HaIR [D( a, Rt] n C[D( a, R), JR.] and h [D( a, R)] C JR.+ , for z in D(a, Rt , h(a)�� I:: S h(z) S h(a)�� ::: .

[Hint: Poisson's formula for h( a) applies to a) , and a) applies to b) .] An echo of the convergence phenomena that obtain for holomorphic

functions is found in


6.2.24 THEOREM. (Harnack) FOR A SEQUENCE {Un}nEN IN Ha(fl) : a) IF un � U ON EACH COMPACT SUBSET OF fl, THEN U E Ha(fl) ; b ) IF {Un}nEN C HaJR (fl) , Un :::; Un+ l , n E N, AND FOR SOME a IN fl, {un(a)}nEN IS BOUNDED, FOR SOME U, un � U ON EACH COMPACT SUBSET OF fl (WHENCE U E HaJR(fl ) ) . PROOF. a) If D( a, r ) C fl Poisson's formula implies U E MVP [D ( a, r)] and so U E Ha (fl) .

b ) The sequence {vn �f Un - Ul } consists of nonnegative functions nEN and Vn :::; Vn+ l , n E N. If b) obtains for {vn } nEN ' for some v , Vn t v and b) obtains for {un}nEN : it suffices to establish b) for {vn }nEN "

If D( c, R) C fl, 0 :::; r < R, and 0 :::; fJ < 27r, then

R - r R + r -- < PI.. (fJ) < -- . R + r - R - R - r Harnack's inequalities, 6.2.23b) , and MVP imply

R - r ( ie) R + r --vn( c) :::; Vn C + re :::; --vn( c), R + r R - r R - r . R + r --v( c) :::; v (c + rete) :::; --v( c) . R + r R - r

The left sides of Harnacks 's inequalities imply that if c E D( c, Rt e fl and lim vn(c) = 00, for all z in D(c, Rt, v(z) = 00. Similarly, the right sides n---+=

of Harnack's inequalities imply that if lim vn ( c) < 00, for all z in D( c, Rt , n---+= v(z) < 00. Those conclusions and the boundedness of {un (a) }nEN imply

0 ¥S �f { Z : z E fl, v(z) < OO } E O(C) , T �f { z : z E fl, v(z) = oo } E O(C) .

Since fl = S u T, S n T = 0, and fl is connected, T = 0: v E MVPJR (fl ) = HaJR (fl) C C(fl, JR.) .

Dini's Theorem (1 .2 .46) implies Vn � v on each compact subset of fl. o

6.2.25 Exercise. For U in HaJR [D(a + ib, rt] : a) if {m , n} C N, then a=+n

a u exists, is harmonic, and there is a power series a=x ny = L cpq (x - a)P (y - b)q p,q=O


converging uniformly on compact subsets of D(a + ib, rt to

b) (Identity Theorem) if

8rn+n 8rnx8ny u(x, y) ;

S C D(a + ib , rt, S· n D(a + ib, rt -j. 0, and u l s= 0, then u ID(a+ib,r) o = O.

[Hint: 6.1.2 applies .]

6.2.26 Exercise. If

{un }nEN c HaIR [D(a, rt] , sup lun (z) l :::; M < 00, I z-a l<r

nEN S c D(a, rt, S· n D(a, rt -j. 0,

and for z in S, lim un(z) exists, then {un} EN converges uniformly on n--+CXJ n compact subsets of D(a, rt.

[Hint: Both 6.1 .2 and 5.3 .60 applY. ]

6.2.27 Exercise. a) In 6.2. 15, if I(z) �f In I z l , then u == O. b) The function 10 3 z r-+ In I z l is harmonic and nonconstant .

6.2.28 THEOREM. IF u E HaIR (10) AND u IS BOUNDED IN 10, FOR SOME U IN HalR (1U) , U lu)= u . PROOF. I f s E JR., in H [D(0.5eis , 0.5tl �f H (fls ) there i s an Is such that Is = u + ivs . If flS I S2 �f flsi n flS2 -j. 0, then � (fs l - IS2 ) I n = 0: i.e. ,

s 1 s2 (fsl - IS2 ) I n is a constant . Thus, for any z in 10,

8 1 82

00 and g is represented by a Laurent series : g(z) = L anzn. n=-(X)

The region fl �f 10 \ ( -1 , 0) is simply connected. For z in fl, there is a rectifiable curve "( : [0, 1] 3 t r-+ fl such that "((0) = 0.5, "((I ) = z. If clef 1 G(z) = 'Y g(w) dw,


then G(z) is independent of the choice of "I and G E H (Q) . For any s , G' - 1� lns = 0, whence G - Is lns i s a constant and

�(G) = �(f )s + constant = u + constant .

clef . clef "0 If a� l = 0' + l(3 and Q 3 z = re' , 0 < r < 1, 1 0 1 < 7r, then

�[G(z)l = u(z) + constant = O'(ln r - ln O.5) - (30

+ '"' [� (an ) cos(n + 1 )0 _ 'S (an ) sin(n + 1 )0] rn+ 1 � n + 1 n + 1 nEZ\{� l }

If r is fixed and (3 i- 0, then limu(z) i- limu(z) , a contradiction since u is Of". O-l-".

continuous in 10. Hence (3 = 0, and there are constants Pn, qn , n E Z, such that

u(z) = O' ln r + L (Pn cos nO + qn sin nO) rn , nEZ

For some M in JR., l u i :::; M and if -n .:s: -1 and P�n i- 0, then

7r [2 (Po + a In r) ± (Pnrn + P�nr�n) 1 :::; 47r M, ( IP�n l r�n + 20' In r) + 2po ± Pnrn �f I + II .:s: 4M.

If a i- 0, then lim I = 00, while I I remains bounded, a contradiction: a = O. dO Thus H �f eG E H (10) , H = constant · eUh, and I h l == 1. Since u is

bounded in 10, so is I H I , and 5.�.11 implies IG I is bounded. Hence 0 is a removable singularity of G. If G denotes the holomorphic extension to 1U of G, then � (C) �f u is harmonic and an extension to 1U of u . 0

[ 6.2.29 No�e. The argument showing that u is the real part of a function G holomorphic near 0 depends on the fact that the original domain of u is a punctured disc. For example, Riemann's Mapping Theorem (8. 1 . 1 ) implies that there is a bijective holo-morphic map I : n+ r-+ Q �f { z : 1�(z) 1 < 1 , "-5(z) E JR.} : in n+ , �(f) is bounded but I I I is not .l


6.3. Subharmonic Functions and Dirichlet 's Problem

The result 6.2.16 is related to a simpler situation for functions of a single real variable. The analog of the equation t:m = 0 for functions u : JR.2 r-+ JR. is f" = 0 for functions f : JR. r-+ R Every such f must be (continuous and) linear, i.e., of the form f : JR. 3 x r-+ ax + b. Furthermore,

f (x; y) == [J(X) ; f(Y)] ,

an identity that can be construed as expressing the MVP in one dimension. Conversely, if f is continuous ,

then g is continuous and g(x + y) == g(x) + g(y). Hence

g(O) = 0, g( -x) = -g(x), g(nx) = ng(x), n E Z, g (rx) = rg(x) , r E Q, g(tx) = tg(x), t E JR.,

g(t) = tg(l ) , f(x) = g(x) - f(O) = g(l )x - f(O) �f ax + b. S· '1 1 h f ' 'd ' . f (x + y) [J(x) + f(y)] 'f Iml ar y, w en IS mz pmnt convex, l .e., -2- � 2 ' 1

{a, ,B} C { t : t = 2kn ' {k, n} e N } and a + ,B = 1, then

f(ax + ,By) :::; af(x) + ,Bf(y) · If, to boot, f is continuous , then f is convex [Ge3] .

A function f in C (JR., JR.) is convex iff either of the following obtains . a) The value of f at the midpoint of every interval does not

exceed the average of the values of f on the boundary of the interval: f (x; y) :::; [J(x) ; f(y)] .

b) For every interval [p, q] , if y is a solution of the equation y" = 0 and y(p) = f(p), y(q) = f(q), i.e., the boundary values of f and y coincide, then for x on [p, q] ,

f(x) :::; y(x) .

When Q c C, for a continuous function u : Q r-+ JR., the following modifications provide analogs of a) and b) above. Intervals [p, q] are replaced

Section 6.3. Subharmonic Functions and Dirichlet's Problem 285

by closed subdiscs of fl. The differential equation y" = 0 is replaced the Laplacian equation t:m = O. Averages of the boundary values on an interval are replaced by averages of boundary values on the boundary of the disc. Below are the precise analogs a' ) resp. b' ) for a) resp. b) .

a' ) For every a in fl and every disc D( a , R) contained in fl, the value of u( a) does not exceed the average of the values of u on CR( a) :

1 12". u(a) :S: - u (a + ReiO ) de. 27r 0 (6.3. 1 )

b' ) For every closed subdisc D( a, R) of fl, if l1v = 0 on D( a, Rt, i .e., if v E HaIR [D(a, rt] ' and if v ICa (R) = u ICa (R) ' then u ID(a ,R) :S: v ID(a ,R) '

As 6.3.5 below reveals, these interpretations, carefully formulated, are equivalent .

6.3.2 DEFINITION. A CONTINUOUS FUNCTION u : C r-+ JR. FOR WHICH a' ) OBTAINS IS subharmonic. WHEN THE DOMAIN OF DEFINITION OF u IS A REGION fl, AND FOR EVERY D(a, R) CONTAINED IN fl, (6.3. 1 ) OBTAINS, THEN u IS subharmonic in fl: u E SH(fl) .

There is an intimate connection, developed below, between certain families of functions in SH(fl) and solutions of Dirichlet 's problem:

For a region fl and an f in C (800 (fl), JR.) , to find a function u such that: a) u E C (flC , JR.) ; b) u E HalR (fl); and c) u l a� (n)= f· A region fl such that Dirichlet 's problem has a solution for every f in

C (800 (fl), JR.) is a Dirichlet region . 6.3 .3 Exercise. I f 0 :s: r :s: 00, the disc D ( a, r t is a Dirichlet region. (When r = 0 the disc is 0; when r = 00 the disc is C. )

[Hint: If 0 < r < 00, Section 6.2 applies ; C (00, JR.) consists of constants.] Ramifications of Dirichlet 's problem are central in the classification of

Riemann surfaces (v. Chapter 10) .

6.3.4 Exercise. a) HalR (fl) c SH (fl); b) SH (fl) is closed with respect to addition and multiplication by nonnegative real numbers and closed with respect to maximization of finite sets:

{Ui E SH (fl) , 1 :s: i :s: I} ::::} {U l V · . . V UI E SH(fl) } .

The equivalence of a' ) and b') above is proved and extended in the next result , which provides alternative views of subharmonicity.


6.3.5 LEMMA. IF u E C (C, JR.) THE FOLLOWING STATEMENTS ARE LOGICALLY EQUIVALENT FOR EVERY DISC D(a, rr. a) u E SH [D(a , Rr] . b) FOR EVERY v IN HaIR [D( a, RrL u - v IS CONSTANT OR FAILS TO

ACHIEVE A MAXIMUM (ON D( a, Rr ) . c ) FOR Ua,R �f p (u I Ca (R) ) ' cf. 6.2.3, U :S; Ua,R ON D(a, Rr .

[ 6.3.6 Note. The function Ua,R is the Poisson modification of u. ]

PROOF. a) ::::} b) : If U - v is not constant on D( a, Rr and

M �f max { u(z) - v (z) : z E D(a, Rr } ' SM �f { z : z E D(a, Rt, u (z) - v(z) = M } ,

since u - v is continuous, S M is closed in D( a, R) 0 . If

b E SM and D(b, rt c D(a, Rr, since v E MVPIR (D(b, r)O ) ,

1 12". M = u(b) - v(b) :s; 27r 0 [u (b + reiO) - v (b + reiO) ] de :s; M.

Hence (u - v) l cb (r)= M. Because r is arbitrary in [0, R - I b - a l ) ,

whence D(b, R - Ib - a i r c SM and so SM is open. Since D(a, Rt is connected and, by assumption, S M -j. 0, S M = D( a, Rt, i .e., u - v is constant , a contradiction. b)::::}c ) : Since Ua,R E HaIR [D(a, RtL w �f u - ua,R is either constant or fails to achieve a maximum on D( a, R)o .

If w is constant , then w = w( a) = u( a) - u( a) = 0, i.e. , u = Ua,R . If w fails to achieve a maximum on D( a, R)O and

M �f SUp { W(Z) : z E D(a, Rt L there is a sequence, {zn} nEN contained in D( a, R)O , converging to some Zoe in D(a, R) , and such that w (zn ) -+ M as n -+ 00. If I zoo l < R, then w achieves a maximum in D( a, Rt, a contradiction. According to 6.2. 15, w ICa (R)= o. Hence, if I zoo l = R, then M = w (zoo ) = 0 as claimed.


c)::::}a) : Because Ua,R is harmonic (whence Ua,R E MVPjR [D(a, Rr] ) ,

Hence u (a) - ua,R(a ) :::; 0, i.e. ,

1 12". u( a) :::; Ua,R( a) = 27r 0 Ua,R (a + ReiO) de 1 12". = - U (a + ReiO) de. 27r 0 o

6.3.7 Exercise. (Maximum Principle in C for functions in SH (Q) ) If a E Q, U E SH (Q) , and u (a) 2: sup u (z ) , then U is a constant function:

zEn u (z ) = u (a) .

6.3.8 LEMMA. (The Maximum Principle in Coo for functions in SH (Q) ) IF {u, -v} c SH(Q) AND, FOR EACH a IN 800 (Q),

inf sup u (z) :::; sup inf v (z) : N(a) EN(a) zEN(a)nn N(a) EN(a) zEN(a)nn

a) u ln< v l n ; OR b) u = v E HajR (Q) .

PROOF. An argument similar t o that in the PROOF for 6.1 .11 applies.

[ 6.3.9 Remark. A function such as v above is superharmonic, i.e., -v is subharmonic. ]

The genesis of the Dirichlet 's problem is to be found in physics. An electrostatic force vector E, arising from the presence of electric charges distributed in ffi.3 , is the gradient of a scalar potential function <t> : ffi.3 3 (x, y, z) r-+ <t>(x, y, z) E ffi. that measures the work required to bring a unit charge from infinity to (x, y, Z) : clef ( E = grad <t> = <t>x, <t>y , <t>z ) . The divergence of E measures the charge density at a point :

div E = div grad <t> = <t>xx + <t>yy + <t>zz '!gf b.<t>.

If the charge is distributed over the boundary 8(Q) of a region Q in ffi.3 , then b.<t> = 0 in Q. The charge distributed over 8(Q) determines <t> I B(n) �f F. Dirichlet 's problem is thus to determine

o


<t> in Q from the knowledge of <t> on 8(Q), i .e., to solve the boundary value problem b.<t> ln= 0, <t> l a(n)= F. The two-dimensional problem can be interpreted as that of finding an elastic membrane described geometrically by the conditions u = u(x, y), (x, y) E Q and statically by the requirement of equilibrium. The system b.u �f uxx + Uyy = 0, u l a(n)= f is the associated differential equation [Ab] . If the topological properties of Q are adequately restricted, Q is a

Dirichlet region. However, not every region is a Dirichlet region, v. 6.3 .11 , 6.3.26, and the observations preceding 6.3.30.

6.3 . 10 Example. If Q = D(O, It and f E C('][', ffi.) , Poisson's formula (v. 6.2.5 and 6.2.15) provides a solution of Dirichlet 's problem .

. clef 6.3 . 11 LEMMA. THE REGION 1U = 1U \ {O} IS NOT A DIRICHLET REGION.

PROOF. Since r � 8(1U) = {O} U '][', if

u E HaIR (10) , f(O) = 1, f l ']['= 0, a E r, and lim u(z) = f(a) , z---+a

then lu i is bounded in 10. Hence 6.2.28 implies that for some U in HalR (1U) , Ulv= u. The Maximum Principle implies that U == 1. There results the contradiction:

1 = lim U(z) = lim u(z) = O. I z itl I z itl

o

For a Dirichlet region Q, a fundamental technique, due to Perron, provides a solution to Dirichlet's problem.

6.3 .12 DEFINITION . FOR A REGION Q, ITS BOUNDARY r �f 8oo(Q) , AND A BOUNDED FUNCTION f DEFINED ON r, THE CORRESPONDING Perron f-family IS F(Q, f) �f { v : v E SH(Q) , lim v(z ) :::; fh) } .

z---+,),Er

6.3 .13 Exercise. For any Q and any f as described in 6.3 .12, the Perron f-family F(Q, f) is not empty.

[Hint: If M = sup I fh) 1 and u(z) = -M, then u E F(Q, f) .] ,),Er

6.3.14 Exercise. For f as in 6.3 .11 , F (10, f) = {O}.

6.3 .15 THEOREM. IF Uf �f sup { u : u E F(Q, f) } , THEN Uf E HalR (Q) . [ 6.3.16 Remark. No condition, e.g. , continuity, etc. , save boundedness is imposed on f.]


PROOF. If M = sup I lh) l , U E F(n, I) , a E 0., and u (a ) > M, 6.3.7 ap,),Er plies when u (z ) == M and leads to a contradiction. Hence u l n:s: M.

If r > 0 and b E D(a, r) c 0., for some sequence {un}nEN in F(n, I) , lim un (b) = Uf(b) . The sequence {wn �f Ul V . . , V Un} is part of n---+= nEN

F(n, I) . Furthermore, the functions

w (z ) �f { P (wn ) (z ) if z E D(a, r)O N n wn(z) otherwise ' n E ,

are in F(n, I) , are harmonic in D(a, rt, and

Harnack's Theorem (6.2.24) implies that if z E D(a, rt,

exists and Wb E HaIR [D(a, rt] ' whence Wb (b) = Uf (b) . Moreover, by defi-nition,

(6.3. 17)

If C E D( a, r t, F(n, I) contains a sequence {un} nEN such that

f clef - d I Yn = Un V Un, n E N, an

Yn(Z) �f { P (Yn ) (z ) if Z E D( a, r)O N Yn (z) otherwise ' n E ,

as in the preceding paragraph, if Z E D( a, r t , then Ye (z ) �f lim Yn (z ) ex-n---+= ists and Ye E HaIR [D(a, rtL whence Ye(c) = Uf (c) . The analog of (6.3.17) IS

(6.3.18)

which is valid by virtue of the definition of the Yn' Hence

Since Wb :s: Ye, V �f Ub,T - Ue,T :s: 0 and so V(b) = O. The Maximum Principle implies VI D(a,T) O== 0, i.e., Wb(c) = Uf (c) . Since c is any point in D(a, rt, Uf E HaIR [D(a, rtl ; since D(a, rt is any open subdisc of 0., Uf E HalR (n). 0


6.3.19 Exercise. The function Uf (the Perron function associated with Q and f) is unique.

Via 6.3.15 there can be constructed a function Uf harmonic in Q and, by abuse of language, majorized by f near r �f 8oo (Q) . Since r is compact, the condition that I f I be bounded is satisfied if f is continuous.

6.3.20 Exercise. A Perron f-family F �f F(Q, f) is a Perron family, viz . , a set F contained in SH(Q) and such that:

{{v E F} 1\ {D(a, Rr c Q}} ::::} {Va,R E F} , {{Vl , V2 } c F} ::::} {:3 (V3 ) {{V3 E F} 1\ {V3 2: max {vl , v2 }}}} '

(6.3.21 ) (6.3.22)

clef 6.3.23 THEOREM. IF F IS A PERRON FAMILY, V = sup v, AND V < 00, F

THEN V E HalR (Q) .

PROOF. The condition (6.3.22) permits the replacement of convergent sequences selected from F by monotonely increasing sequences in F. Hence Harnack's Theorem is applicable. An argument similar to that in the PROOF of 6.3 .15 applies . 0

Absent topological conditions on Q, Q can fail to be a Dirichlet region, e.g., if Q = 10 � 1U \ {O}, v. 6.3 .1l . When Q = 10 the following are noteworthy features:

• r �f 8oo (Q) = {0}l:J1I'; • HaIR (Q) n C (Qc , JR.) contains no function (3 such that

(3(z) is { positive �f z E r \ {O} (= 1I') . zero If z = 0

(Indeed, for such a (putative) (3, P ((31 r) �f v E HaIR (1U) n C (U , JR.) . Since (3 is positive on 1I' and v E MVPIR(1U) , v(O) = v(O) - (3(0) > O . If, for some a in 10, v (a) - (3(a) �f E > 0, for z in 1Uc and

p(z) �f v (z) - (3(z) + E l z l 2 ,

p lT= E and p(a) > Eo Hence for some b in 1U, p(b) 2: maxp(z) . Conse-zEl[JC

quently Px (b) = py (b) = 0 and Pxx (b) < 0, pyy (b) < 0: t:J.p(b) < O. On the other hand, since v - (3 is harmonic in 10, t:J.p = 4E > 0, a contradiction. Thus v - (3110= 0 and lim v (z) - (3(z) is both v(O) (> 0) and,

z--+o by virtue of continuity, 0, a contradiction: no (3 as described exists. )

The preceding considerations motivate


6.3.24 DEFINITION. FOR A REGION Q AND AN a IN r �f 800 (Q) , A FUNCTION (3 IN HaIR (Q) n C (Qc , JR.) SUCH THAT

IS A barrier at a.

(3(z) is { positive IF Z E (r \ {a}) zero IF Z = a

6.3.25 Exercise. If Q is a Dirichlet region, there is a barrier at each point a of 800 (Q) .

6.3.26 Exercise. If Q �f 1U \ { -� , � } there is no barrier at either -� 1 or 2 . (Hence Q is not a Dirichlet region. ) For 6.3.25 there is a converse derived from

6.3.27 THEOREM. IF Q IS A REGION, a E 800 (Q) �f r , AND THERE IS A BARRIER AT a, FOR EACH f IN C(r , JR.) AND THE ASSOCIATED UNIQUE PERRON FUNCTION UJ , lim UJ(z) = f(a) . z E n

PROOF. If E > 0, for some positive r, Z E D( a, r t n r implies

I f(z) - f(a) 1 < E.

The hypothesis implies that (3 has a positive minimum m on the compact set r \ D(a, rt . If M �f I l f l loo the function

u : QC 3 Z r-+ f(a) + E + (3(z) [M - f(a)]

m

is in HaIR (Q) , and

u(Z) > { f(a) + E > f(z) if z E [D(a, rt n r] - M + E > f(z) if z E { [C \ D(a, rt] n r} .

If v E F(Q, f) , by virtue of the Maximum Principle in Coo ,

whence (UJ - u) In:::; o. Therefore,

inf sup UJ(z) :::; u(a) = f(a) + E. N(a) EN(a) zEN (a)nn

On the other hand, the function

(3(z) w : QC 3 z r-+ f ( a) - E - - [M + f ( a )]

m


is in HaIR (Q) , and

w(z) < { J(a) - E < J(z) - -M - E < J(z)

if z E [D(a, r)O n r] if z E { [C \ D(a, r)O] n r} .

Hence the harmonic function w is in F(Q, J) and so (UJ - w) I n2': 0, i.e . , sup inf UJ(z) 2': w(a) = J(a) - E. 0

N(a) EN(a) zEN (a)nn

6.3.28 COROLLARY. IF THERE IS A BARRIER AT EACH a IN r, FOR EACH J IN C(r, JR.) , DIRICHLET'S PROBLEM HAS A UNIQUE SOLUTION.

PROOF. The result 6.3.27 applies. [ 6.3.29 Note. Aside from useless tautologies, there seems to be no general necessary and sufficient condition for the existence of a barrier at a point a in 800 (Q) .

On the other hand, a sufficient condition for the existence of a barrier at a in 800 (Q) is the following.

If Q, a E 800 (Q) , and the component of Coo \ Q that contains a is not a itself, there is a barrier at a, v. 8.6.16.]

o

The following observations provide some orientation about Dirichlet regions and non-Dirichlet regions.

If Q is a simply connected proper subregion of C, then Q is a Dirichlet region. (Owing to 8 .1 .8d) , Coo \ Q consists of one component and is not a single point. )

Although 800 (q = 00 (a single point) , nevertheless C is a Dirichlet region. (The (constant) function u == J(oo) is a solution of Dirichlet 's problem.

If 0 < r < R :::; 00, the region A( a; r, R) 0 , which is not simply connected, is a Dirichlet region.

The regions 10 (= A(O; 0, It) and C \ {O} are not Dirichlet regions, v. 6.3. 11 .

6.3.30 Exercise. I f r > 0 , D(b, rt n Q = 0, and a E { [800 (Q)] n Cb(r) } ,

then (3 : QC 3 z r-+ In (�) - In I z -a; b I is a barrier at a.

Alternative definitions of the notion of a barrier at a m r �f 800 (Q) are:

Section 6.3. Subharmonic Functions and Dirichlet 's Problem 293

a) ( [Re, Ts] ) A barrier at a is a function (3 defined in some open neighborhood N(a) , continuous on (N(a) n Q)C , subharmonic on N(a) n Q, and such that

(3(Z) is { positive �f z E [(N(a) n Q)C \ {a}] . zero If Z = a

b) ( [Con] ) A barrier at a is a family {(3r }r>O such that: a) (3r is superharmonic in Q n D(a, rt and 0 :::; (3r (z) :::; 1 ; b) lim (3r (z) = 0; c) if

z---+a w E Q n Ca (r) , then lim (3r (z) = l . z---+w

c) ( [AhS, Be] ) A barrier at a is a function (3 such that: a) (3 E SH (Q) ; b) lim (3(z) = 0; c) if b E 8(Q) \ {a} , then lim(3(z) < o. z--+a z=b

6.3.31 Exercise. If Q is a Dirichlet region, for each a in 800 (Q) , there is, in the sense a) or b) a barrier at a.

6.3.32 Exercise. If, in the sense a) , (3 is a barrier at a, then

lim UJ(z) = f(a) . z---+a

6.3.33 Exercise. If, in the sense b) , (3 is a barrier at a, then

lim UJ (z) = f(a) . z---+a

6.3.34 Exercise. If, in the sense c ) , (3 is a barrier at a, then at a there is a barrier as defined in 6.3.24.

[Hint: When f = (3, 6.3.15 applies followed by a change of sign.] For some purposes the discussion of subharmonicity is carried out in

the following more general context.

6.3.35 DEFINITION. A FUNCTION u IN JR.r1 IS subharmonic in the wide sense , i .e. , U E SHW(Q) , IFF: a) u(Q) C [-00, (0) ; b) u E usc (FOR A SEQUENCE {Un}nEN OF CONTINUOUS FUNCTIONS, un ..l- U, v. 1 .7.24) ; c) FOR EVERY a AND EVERY POSITIVE R SUCH THAT D( a, Rt C Q,

1 1271" u(a) :::; - Un (a + ReiO) dO, n E N. 27r 0 (6.3.36)

6.3.37 THEOREM. IF U E usc(Q) AND u(Q) c [-00, (0) , u E SHW(Q) IFF FOR EVERY COMPACT SUBSET K OF Q,

{ {v E HaIR (KO ) n C(K, JR.) } 1\ { v l a(K) 2: U l a(K) } } ::::} {v i K2: u l K } ' (6.3.38)


i.e. , IFF DOMINATION ON 8(K) OF U BY A HARMONIC FUNCTION v IMPLIES DOMINATION OF U ON K BY THE SAME v .

PROOF. If U E USC(Q), U (Q) C [-00 , (0) , and U E SHW(Q) , the averaging argument used for 6.1.8 , 6. 1 . 11 , 6 .1 .12 , and 6.3.8 yields, for each compact subset K of Q and each v in HaIR(Q), the inequality (6.3.36) .

For the converse, if r > 0 and D( a, r) 0 C Q, for some sequence {un} nEN of functions continuous on D( a, r) , Un ..l- u. Thus

Thus U E SHW(Q) .

{Vn �f P (Un) } c HaIR [D(a, rtl , nEN

6.3.39 THEOREM. (Hartogs) IF: a) {un}nEN C SHW(Q) , AND b)

{Z E Q} ::::} { lim Un (z) � C < oo} , n---+=

o

FOR EACH COMPACT SUBSET K OF Q AND EACH POSITIVE E THERE IS AN no(K, E) SUCH THAT ON K

(6.3.40) (For the significance of a) , v. 6.3.42) .

PROOF. For some M and all n, Un � M on K since otherwise, K contains a sequence {zn}nEN such that for some Z= in K and some sequence {mn}nEN ' mn 2': n, lim Zn = Z= , UTnn (zn) > n. Then, because each Un is usc, n---+=

lim Un (z=) = 00 (> C) , n---+= a contradiction. Hence, for the purposes of the argument, the assumption Un I K < 0 is admissible. Since Un is usc,

clef { K c U = z : un(z) < o } E 0(((:) , whence un l u< o.


Because K is compact , 1 .2.36 implies that for some fixed positive r, U D(z, 2rt C U. If Z is fixed in K and un(z) > -00, since Un E SHW(Q), zEK

Since un lu < 0, if v E U and I z - v i < rS < r, then replacing r by r + rS in the last inequality leads to

7r(r + rS) 2un (v) :::; r un(w) dA2 (W) JD(v,r+W

:::; r un(w) dA2 (W) . J D(v,r) O

(6.3.41 )

On the other hand, (6.3.40) implies that if E is a measurable subset of D(z, 2rt , then lim r un(w) dA2 :::; r lim un(w) dAs :::; CA2(E) . Thus,

n�= JE h n�=

for some nz , if n > nz , then Ie un(w) dA2 < (C + E) . A2 (E) .

Consequently, if rS is sufficiently small (and positive) and n > nz , then

and (6.3.41) implies that if n > nz , then un(z) < C + E. Since r is z-free,

for some finite set Z1 , . · · , zm , K C U D (zm , rSt . If k= 1

( ) clef nO K, E = max nZk 1�k�rn the desired conclusion follows.

E 1 clef [ ] 2 ("\ clef 6.3.42 xamp e. If K = 0, 1 , H = C,

if x > O , if x :::; °

'f 1 1 X < -n 1 ' if x 2': -n

o


Un E COO (C, lR.), lim Un == 0, but Un ( 2� ' Y) = n. n---+oo

Hence on K, {un }nEN is unbounded. The condition (6.3.36) serves to control the behavior of the sequence so that its pointwise boundedness is strengthened to uniform boundedness on compact sets.

6.4. Appendix: Approximate Identities

Fejer's kernels (3.7.6 and 4.10. 17) , the functions Uv (4. 10.13) , and the Poisson kernels (Section 6.2) constitute sets of functions that are examples of approximate identities as described below.

The context for an approximate identity is a topological algebm, i.e. , an algebm A over a topological field K and endowed with a Hausdorff topology T. The maps A x A 3 (x, y) r-+ xy, A x A 3 (x, y) r-+ x - y, and K x A 3 (a , x) r-+ ax are assumed to be continuous.

6.4.1 Exercise. Every Banach algebra is a topological algebra. In a topological algebra A a net n : A 3 ), r-+ n;. E A is an approximate

identity iff, for each x in A, n;. . x converges to x. 6.4.2 Exercise. For some topological algebra A resp. B the Poisson kernels resp. Fejer kernels constitute an approximate identity.

6.4.3 Exercise. If A is a commutative Banach algebra and

n : A 3 ), r-+ n;. E A

is an approximate identity, ii:)..(M) � l .

6.4.4 Exercise. a ) I f G i s a locally compact abelian group, p, i s Haar measure, and A �f L 1 (G, p,) , and for each nonempty open neighborhood N of h ·d · clef 1 h { } . · ·d t e 1 entlty e , nN = p,(N) XN , t en nN nEN(e) IS an�proxlmate 1 en-

tity. b) For any approximate identity {n;.hEA in A, (n;' ) [x] (M) -+ aM(x) and the convergence is uniform with respect to the parameter x.

6 . 5 . Miscellaneous Exercises

6.5.1 Exercise. If f E Ha(Q) and 0 tJ- f(Q), then In I f I E HaJR (Q) . 1 [Hint: If f = U + iv, then If I = (u2 + v2) 2" .J


6.5.2 Exercise. If Q is star-shaped and u E HaIR (Q) , some v is a harmonic conjugate of u (in Q).

6.5.3 Exercise. If u has continuous partial derivatives of the second order in Q and l1u > 0 resp. l1u < 0 in Q, then u is subharmonic resp. superharmonic in Q; u is subharmonic in Q iff l1u 2: 0 in Q. 6.5.4 Exercise. If

then u E HaIR (C= ) .

clef { �(z) u(z) =

0l z l 2

if z -j. 0 otherwise

6.5.5 Exercise. Which of 6.3.4, 6.3.5, 6.3.10, 6.3.13, 6.3.14, 6.3.19, 6.3.23

remains valid and/or meaningful when subharmonic is replaced by subharmonic in the wide sense? 6.5.6 Exercise. If 0 < r < 1 and u E HaIR [A(O; 0, It] : a)

1 *du = 0 I z l=r

(v. Section 5.8 ) ; b) for Zo in A(O; 0, It, the map

F : 1U 3 z r-+ u (zo ) + lz (du + i * du)

Zo

(cf. Section 5.8 ) is well-defined and F E H [A(O; 0, It] . [Hint: For a), Stokes's Theorem applies, and a) applies for b ) .] The next two results are used in Chapter 10.

6.5.7 Exercise. If f E H [A(O; 0, 2t] and 0 < p < 1 : a) for some up in HaIR [A(0; p, 2t] ' up I 1 z l=p= �(f) l l z l=p ; b) if P < I zo l , I z l < 2 and

(v. 6.5.6b) ) , then Fp - f is represented by a Laurent series L cnzn; c) if nEZ

p < s < 2, there are sequences {an} nEZ and {bn } nEZ such that

[Up - �(f)] (reiO ) = L (an cos n() + bn sin n(}) sn ; nEZ


d) for m in Z+ ,

� t [up - �(f)] (reiO) cos mO dO = amsm + LmS�m , 7r Jo � t [up - �(f)] (reiO) sin mO dO = bmsm + b�ms�m ; 7r Jo

e) when s = p resp. s = 1 , then a�m = _p2mam and b�m = _p2mbrn resp. for Mp �f max lup l + max 1 �(f) I ,

I z l=l I z l=l

. 2 f) If 0 < p < 1 , max { I am l , Ibm l } :::; --2Mp; g) mEN+ 1 - P

h) max lup l :::; max 1 �(f) 1 + 8Mp_r_; i ) if 0 < r < � , then I z l=l I Z I=r 1 - r 5

1 - r ( ) Mp :::; -- max 1 �(f) 1 + max 1 �(f) 1 : 1 - 5r I z l=r I z l=l

Conclusion: If 0 < r < � , for some function k(r) , 5

[Hint: For h) the Maximum Principle and f) apply.]

6.5.8 Exercise. If K(C) 3 K and fl �f C \ K is a Dirichlet region, in C (flc , lR.) there is a u such that: a) u l a(fl) = 1 ; b) u E HaJR(fl); c) in fl, O < u < 1 .

[Hint: a) For some nonconstant superharmonic v, v lK= 1 and at some a in fl, v(a) < 1. b)


is a nonempty Perron family. c) W �f sup w E HaIR (Q) . 4. For F

some relatively compact Dirichlet region Q 1 , K c Q 1 . 5. If g is a (harmonic) solution of Dirichlet 's problem { I if z E K g(z) = 0 if z tJ- Ql '

then g - v E SH (C) and g - v l lC\n:::; O. 5. g E F, g :::; W :::; v, and W serves for u.]

299

6.5.9 Exercise. a) If K is compact , then Q �f C \ K is not relatively compact . b) The set of 0 of Dirichlet regions Q, that are relatively compact and contain K is nonempty. c) If 1 E HaIR (Q) , 0 :::; I :::; M < 00, for each t , there are solutions u, and v, for Dirichlet 's problems:

{u" vd c HaIR (Q, n Q) ,

( ) _ { I (z) if z E 8( K) u, z - 0 if z tJ- Q, ' and

( ) = { I if z E 8( K) v, z 0 if z tJ- Q, .

d) U �f sup u, E HaIR (QC) and V �f sup v, E HaIR (QC) . e) o 0

6.5.10 Exercise. If Q is convex and for each z in Q, � [I' (z) ] -j. 0, then 1 is injective on Q.

[Hint: If

then

1 clef . { clef .(3 clef .S:} n = U + lV , a = a + l , C = "I + l u C H,

C - a = "1 - a + i (J - (3) �f P + iq, t E [0 , 1] ' F(t) �f 1[(1 - t)a + tc] �f u[x(t) , y(t)] + iv [x(t) , y(t)] , c - a p + lq

(p2 + q2) F(t) = pu[x(t) , y(t)] + qv [x(t) , y(t)] + i {pv [x(t) , y(t)] - qu[x(t ) , y(t)] } .


For some T and a in [0, 1] , � �f X(T) + iY(T) and 1] �f x(a) + iy(a) ,

= pUx [X(T) , Y(T)]p + pUy [X(T) , Y(T)]q + qVx [X(T) , Y(T)]p + qVy [X(T) , Y(T)]q + i {pvx [(x(a) , y(a)]p + pvy [x(a) , y(a)]q} - i {qux [(x(a) , y(a)]p + quy ( [x(a) , y(a)]q} .

Owing to the Cauchy-Riemann equations, the right member of the display above is (p2 + q2 ) {� [f'(�)] + is' [f' (1])] }. ]

6.5. 11 Exercise. If z �f rei</> E 1U, E E S,B ClI') , for the harmonic measure w(z , E) �f � r Pr((} - ¢) d(} of E with respect to z: 27r JE

a) w(O, E) = T(E) ; b) lim w (z, E) � { I r---+ 1 0

clef " 0 clef " [Hint: If w = e" , w' = e'w , z E [W, w'] , then

1 - Iz l 2 = Iw - z l ' Iw' - z l , dw

IW' - z l

if ei</> E E otherwise

d(}

Iw - z l dw

{ ,B and

d(} = Pr. ((} - ¢) . If E = e" o :::; (31 :::; (3 :::; (32 :::; 27r } , the

results are valid.]

7 Meromorphic and Entire Functions

7.1. Approximations and Representations

A locally holomorphic function I is defined in some Q throughout which it is differentiable. When Q = C, I is entire : l E E. The results for entire functions have an importance that is reflected in the general theory of locally holomorphic functions, e.g. , in the study of Blaschke products for representing functions I in H (1U) .

If l E E and zut -j. 0, the Identity Theorem 5.3.52 implies 1 == 0. Thus zut = { C (and 1 == 0)

. ° and I =t=- 0 I' Consequently, if I =t=- 0, then f E M(C) , and the study of E is thereby

related to the study of M(C) . The dominating theorem for M(C) is that of Mittag-Leffler. His result is a consequence of a later and more general result of Runge. Thus, for concision and efficiency, Runge's Theorem is the first object of study in the current Chapter.

1 For a in C, the function Ra : C \ {a} 3 Z r-+ -- is an example of z - a the simplest type of meromorphic function. Not unexpectedly, a study of Ra leads to a more refined understanding of more general meromorphic

00 functions, e.g. , rational functions, convergent series L rn (z) in which each

n=l

term is a rational function and [U P (rn )] · = 0, etc. nEN

The following observation is reminiscent of the argument used in de-riving a power series representation via Cauchy's integral formula.

If r > 0, w E D(a, rt , and I z - a l 2: r, then

1 Rw (z) = z - a - (w - a) ( 1 ) � (w - a)n = Ra (z)

w - a = � (z _ a)n+l . 1 - -- n=O z - a

(7. 1 . 1 )

301

302 Chapter 7. Meromorphic and Entire Functions

Convergence of the right member of (7. 1 . 1 ) is uniform on the closed set { z : I z - a l 2: r } .

What follows exploits these elementary conclusions to reach a considerable generalization.

The global version of Cauchy's integral formula (5.4.14) may be viewed as a special case of the next result , which is the key to the derivation of Runge's Theorem.

7. 1 .2 THEOREM. IF (O(C) \ 0) 3 U ::J K E K(C) , THERE ARE RECTIFIN ABLE CLOSED CURVES bn h<;n<;N SUCH THAT: A ) U 1'� C U \ K; B ) IF

J E H (U) AND z E K, THEN

N J(z) = � L 1 J(w) dw. 27rl � W - Z n= l I n

n=1

(7.1 .3)

[ 7.1 .4 Remark. As the proof below reveals, each 1';' is the union of a finite number of horizontal and vertical complex intervals.]

PROOF. The argument is conducted in terms of closed squares

clef [ p p + 1 ] [ q q + 1 ] Qpqrn = 2m ' � X 2m ' � , {p, q, m} C Z.

The oriented boundary aOQpqm of Qpqm is the union of four oriented sides :

and if

l'(t) =

a + 4(b - a)t b + 4(C - b) (t - �) C + 4(d - C) (t - �) d + 4( a - d) (t - �)

if 0 < t < � - - 4 . 1 1 If - < t < -4 - - 2

1 3 ' if - < t < -2 - - 4 .f 3 1 - < t < l 4 - -

Section 7.1 . Approximations and Representations 303

Since K is compact (hence bounded) , for any m, K is covered by a set Q consisting of finitely many Q . . . each of which meets K. There is a second set S consisting of finitely many Q . . . not in Q but meeting U Q .. . . The set

Q

K �f ( U Q .. . ) is compact and if m is such that Qus

3 . - < mf { l z - w l 2m z E K, w E (C \ U) } ,

then K c KO C U. A side belonging to only one constituent square of K is unshared; all

other sides of constituent squares of K are shared. If s is shared by Q . . . and Q . . . and, as an oriented side of Q . . . , s = [a, b] as in Figure 7.1 . 1 , then as an oriented side of Q . . . , s = [b, a] . Furthermore, for any g,

r g dz Jra,b]

exists iff r g dz exists and r g dz = - r g dz. The oriented bound-J�,a] Jra,� J�,a]

ary r �f 8° K is a union of horizontal and vertical oriented complex closed intervals that are unshared sides of constituent squares Q . . . of K. Any configuration of intersecting unshared sides is, modulo rotations through 7r integral multiples of "2 radians, one of those illustrated in Figure 7. 1 .1 .

w w u

Figure 7.1 .1 . Solid lines are un shared sides of constituent squares of K. 'Dotted lines are shared sides of constituent squares of K.


A vertex v (an endpoint) of an unshared side is the intersection of two or four unshared sides. Accordingly, v is a 2-vertex Or a 4-vertex. As indicated in Figure 7.1 .1 , for a given v, there are in r algorithmically defined vertices u and w such that { [u , v] , [v, w] } are oriented, with respect to the squares to which they belong, according to the pattern in (7 .1 .5) .

Thus, if v? is a vertex in r there is in r an algorithmically defined vertex vt such that [v? , vt ] is oriented according to the pattern in (7. 1 .5) and an algorithmically defined vertex v� 1 such that [v� 1 , v?l is oriented according to the pattern in (7. 1 .5 ) . By induction there is generated a sequence

of algorithmically defined vertices to which no further algorithmically defined vertices can be adjoined: 0"1 is maximal.

An examination of the possibilities: a) -Pl VI is a 2-vertex and vil is a 2-vertex; b) -Pl VI is a 2-vertex and vil is a 4-vertex; c) -Pl VI is a 4-vertex and vi1 is a 2-vertex; d) -Pl VI is a 4-vertex and vi1 is a 4-vertex;

leads to the conclusion: Only a) or d) can be valid.

Owing to the maximality of 0"1 , a) implies V�Pl = vil ; d) does not imply V�Pl = vil However , every vertex in

ql - l r �f U [ j j+l ] 1 - VI ' VI

j=-Pl is either a 2-vertex or a 4-vertex, whence r 1 is an oriented cycle (v. Section 5.5) , i.e. , r 1 is a closed curve-image.

If v� E (r \ r d , the procedure just described may be repeated to proq2- 1

duce an oriented cycle, r 2 �f U [v� , V�+I] . After finitely many repetij=-P2

tions of the procedure, all the vertices in r are consumed and there emerge N oriented cycles r 1 , . . . , r N such that r = U rn .

n=1 For each n there is a rectifiable closed curve "In such that "In lin-[ j j + l ] early maps onto the (j + l)st interval of r no Hence Pn + qn ' Pn + qn "I� = r n, 1 :::; n :::; N. If z is on no side of any Q . . . , then

if z E Q�. otherwise (7. 1.6)


Owing to cancellations that result from integration over shared sides, for z as described above and in K,

1 27ri C�JOQ

�� dw + Q�JoQ

��'� dw ) = L Ind aoQ (z)J(z) + L Ind aoQ . . (z)J(z)

Q . . . EQ Q . . ES

= _1 (t 1 lJ:!!l dW) 27ri n=l 'Yn W - Z •

(7. 1 . 7)

(7.1 .8)

Because of (7.1 .6) , the right member of (7. 1 .8) is J(z) . Furthermore, since r n K = 0, the right member of (7. 1 .8) and J(z) are continuous functions

of z on all K. Thus, if z E K, then J(z) = � � 1 J(w) dz. 0 27rl � � W - Z n=l I n

7.1.9 Exercise. The assertions about the possibilities a)-d) are valid. [ 7. 1 .10 Note. As elaborated in [New] , the argument underlying 7.1 .2 leads to Alexander 's proof of the Jordan Curve Theorem.]

7.1 .11 THEOREM. (Runge) IF: a) K 3 K c Q c C; b) 5 c (C= \ K) AND 5 MEETS EACH COMPONENT OF C= \ K; c) J E H (Q) ; THEN J IS UNIFORMLY APPROXIMABLE ON K BY RATIONAL FUNCTIONS R FOR WHICH P(R) C 5.

PROOF. If R is the set of all rational functions r such that P(r) C 5, each such r has only finitely many poles, and thus each such r is in H (K) . Furthermore M �f R I K is a vector subspace of C (K, q . The content of 7.1 .11 is that J (in H(Q) n C(K, q ) is in the I I I I=-closure of M.

If the assertion is false, the Hahn-Banach Theorem and the Riesz Representation Theorem imply there is a measure space (K, S,B , p,) such that

1 J(z) dp,(z) = 1 and {g E M} '* {l g(z) dp,(z) = o} . For this p" h : C= \ K 3 z r-+ r dp,( w) is holomorphic in C= \ K. If C is

iK w - z a component of C= \ K, 00 E C, and 00 i- s E (5 n C), for some positive E, D(S , Et c C. If z E D(S, Et, for w in K,

N " (z - s )n �

_1 _ . � (w - s )n+l W - Z

(7 .1 .12)


Each term R(w) in the left member of (7. 1 .12) is in M, whence

i R(w) df.L(w) = 0,

and the Identity Theorem implies h lc= 0. When s = 00, (7. 1 .12) is replaced by

N n 1 '" w u � zn+l -+ w - Z ' n=O

and convergence obtains throughout K if I z l > sup { Iw l lows that h l lC= \K= 0.

Then 7.1 .2 yields

w E K } . It fol-

1 = r J(z) df.L(z) = r [�t 1 J(w) dW] df.L(v) . } K } K 27rl n=l "In W - Z

Fubini's Theorem permits reversing the order of integration in the right

member above. Since h = 0, 1 = � t 1 J(w)h(w ) dw = 0, a contra-27rl n=l "In

diction. 0 [ 7.1.13 Remark. Since Coo is separable and the components of the open set Coo \ K are open, there are at most count ably many such components. For each component C of Coo \ K, 5 n C may be a single point , i .e . , 5 may consist of at most count ably many points, exactly one in each component of Coo \ K. For such an 5, the available rational functions constitute a minimal set; the impact of Runge's Theorem is maximal.]

7.1 .14 COROLLARY. (Polynomial Runge) IF Coo \ Q IS CONNECTED, POLYNOMIAL FUNCTIONS SERVE AS THE RATIONAL APPROXIMANTS R. PROOF. An admissible choice for 5 is {oo}. If R is a rational function and P(R) C {oo}, then R is entire, i.e. , a polynomial. 0 7.1 .15 Example. a) When Q = D(O, 2t , Kl = D(O, 1 ) , and J E H (KI ) , for some positive E, J is representable by a power series for which the radius

00 of convergence is 1 + E: J( z) = L cnzn . Hence, on K1 , J is uniformly

n=O N approximable by the sequence of polynomials L cnzn, N E N" which are

n=O functions in H [D(O, 2t] .


1 b) On the other hand, if K2 = { z : I z l = I } and I(z) = - when z z -j. 0, then I E H (K2) . If g E H(Q) , then

r g(w) dw = ° while r I(w) dw = 27ri. iK2 iK2

Thus g cannot approximate I uniformly on K2• What is an essential difference between K1 and K2? For the com

ponent e2 �f D(O, It of Q \ K2 , e� n Q = D(O, 1) is compact . However , e1 �f { z : 1 < I z l < 2 } , is the only component of Q \ K1 and

which is not compact : e1 is not relatively compact in Q. Consequently, Runge's Theorem has a number of variants. In the con

text of 7. 1 . 11 , the set R may be replaced by H(Q) . In particular, the following result , called Runge's Theorem by some, applies when K is a compact subset of an open set U.

7.1 .16 THEOREM. (Runge-variant) IF K(C) 3 K C U E O(C) , THE FOLLOWING STATEMENTS ARE EQUIVALENT:

a) IF I E H(K) , I IS UNIFORMLY APPROXIMABLE ON K BY FUNCTIONS IN H (U) ;

b) IF e IS A COMPONENT OF (THE OPEN SET ) U \ K, THEN ec n U IS NOT COMPACT, i.e. , NO COMPoNENT OF U \ K IS RELATIVELY COMPACT IN U;

c) IF z E U \ K, FOR SOME I IN H(U) , I /(z) 1 > sup I /(w) l . wEK

PROOF. The pattern of proof is: c) ::::} b); a) ::::} b) ::::} a) ; b) ::::} c) .

c) ::::} b) : If b) is false. for some component e of U \ K, ec n U is compact . If p E Be, some sequence {Pn } nEN contained in U converges to p. Since e is open, p tJ- e, whence p E ec n U c U, P E U \ e = K: Be c K. The Maximum Modulus Theorem 5.3.36 implies sup I /(z) 1 :s; sup I /(z) l , a denial of c) .Hence

-,b) ::::} -,c)

and c) ::::} b).

Cc K

(7. 1 . 17)

a) ::::} b) : If b) is false and I E H(K) , for some sequence {fn}nEN in H (U), In � I on K. Furthermore (7. 1 . 17) implies that for all n and m in N,


sup I/n (z ) - Irn(z) 1 � sup I /n (w ) - Irn(w) l · Thus {fn}nEN converges uni-zECC wEK formly on Ge to some F. Hence F is continuous on Ge , F = 1 on K, which contains BG (v. preceding argument for c) ::::} b) ) , and F E H(G) . If w E G, there are an open V containing K and an N ( w) such that V n N ( w) = 0.

A particular choice of 1 is given by the formula I(z) �f _1_ . Then z - w

1 E H (K) , (z - w)F(z) = 1 on an open set W containing K, and on the nonempty open set G n W. The Identity Theorem implies (z - w)F(z) = 1 on G. When z = w, there emerges the contradiction, 0 = 1 : a) ::::} b) .

b) ::::} a) : If a) is false, I, viewed as an element of G(K, q , is not in the closure of H (U) viewed as a subset of G(K, q . The Hahn-Banach Theorem (3.3.8) implies there is a measure space (K, S,B , p,) such that for every g in H(U) , iK g dp, = 0, while iK 1 dp, = 1 . The following argument shows that the last equation is false.

The map ¢ : C \ K 3 w r-+ r _1_ dp,(z) is in H(C \ K) (v. 5.3.25). iK z - w

If w E C \ U, since z r-+ l )k 1 is in H(U) , the Hahn-Banach Theorem (z - w + implies here that

¢(kl (W) = k! r (

1 )k 1 dp,(z) = 0, k E Z+ . iK z - W + (7. 1 . 18)

1 00 zn If Iw l > sup I z l , then -- = - '" --1 ' and the series converges uni-K z - W � wn+ n=O formly for z in K. Since i zn dp,(z) = 0, n E N+ , ¢(z) = 0 if z is in the (unique) unbounded component of C \ K.

If V is a bounded (open) component of C \ K and V n (C \ U) = 0, then V C U and the boundedness of V implies Ve n U is compact , a contradiction of b) : V n (C \ U) -j. 0. If w E V n (C \ U) , (7. 1 . 18) implies that for some neighborhood W of w, ¢I w= 0, whence ¢I v= o. In sum, ¢I IC\K = o.

If N is a neighborhood of K, the compactness of K implies that N may be assumed to be the union of finitely many open squares Q�. of the form in 7.1.2 . For some infinitely differentiable '1jJ defined on N, '1jJ == I on K and '1jJ 1 1C\N = 0, in particular, '1jJ liJN = o. If 1 E H (N), al = o. According to the product rule for derivatives, aU · '1jJ) = (af) . '1jJ + I · a'1jJ = I · a'1jJ. Hence Pompeiu's formula (5.8.14) implies that if z E K, then

1 1 B'1jJ(w) -I(z) = l(z)'1jJ(z) = -. I(w) -- dw 1\ dw .

27rl N W - Z

Section 7.1 . Approximations and Representations

Fubini's Theorem and the definition of ¢ imply

r J(z) dpJz) = -� r (1 J(w) 8'1jJ(w) dw 1\ dW) dp,(z) } K 27rl } K N W - Z

= 1 J(w)8'1jJ(w) . ¢(w) dw 1\ dw .

309

Since ¢I IC\K = 0 and 8'1jJ(w) = 0 on K, i J dp, = 0, a contradiction. (A similar technique is employed in the PROOF of Runge's Theorem.)

b) ::::} c ) : If z E U \ K, for some positive r, D(z, r) c U \ K. If C is a component of U \ [K U D(z, r)] ' either C or C U D(z, r) is a component of U \ K. Thus b) obtains for K U D(z, r) . Since K n D(z, r) = 0, there exist disjoint open neighborhoods N(K) and N[D(z, r)] . Hence there is in H {N(K) U N[D(z, r)] } a function J such that J I K= 0 = 1 - J I D(z ,T) ' Since b) ::::} a) , there is in H (U) a g such that

from which c) follows. o 7.1.19 Exercise. If K is compact , every function in H (K) is uniformly approximable on K by polynomials iff C \ K is connected.

[Hint: a) In the current context, for U in 7. 1 .16, C may serve. b) A component of C \ K is relatively compact in C iff C \ K is not connected. c) If J is entire, J is uniformly approximable on each compact set by polynomials.] The similarity of the techniques used in the argument for Runge's

Theorem and its variant leads, by abuse of language, to the conclusion:

Runge-variant ::::} Runge.

7.1 .20 Exercise. For the set K in the argument for 7. 1 .2 , no component of U \ K is relatively compact in U. (The condition 7.1 .16b) may be interpreted roughly as saying that part of the boundary of each component of U \ K meets the boundary of U, v. 7.1. 15b) . )

Since the components of Coo \ K play a role in the previous discussions, the following result is of interest , and proves central in 8.1 .8 .

7. 1 .21 THEOREM. IF Q IS SIMPLY CONNECTED, THEN F �f Coo \ Q IS CONNECTED.


PROOF. As a closed subset of Coo , if F is not connected, there are two disjoint , closed, and nonempty sets J and K such that F = Jl:JK; 00 is in one, say J, and thus K is compact . Furthermore O(q 3 U �f C \ J = Ql:JK. If f == 1 in 7. 1 .2 , one of the summands, e.g. , 1 _

1_ dw, in (7 .1 .3 ) is

/'1 w - Z not zero. On the other hand, U \ K = (C \ J) \ K = (Ql:JK) \ K = Q and if Z E K, then _

1_ , as a function of w , is holomorphic in Q . According

w - Z to 5.3.14b) , 1 _

1_ dw = 0, a contradiction. 0 /'1 W - Z

7.1.22 Example. The complement (in q of a nonempty compact subset of C is not simply connected. The region C \ (-00 , 0] is simply connected. Although the complement (in q of the strip S �f { z : -7r < 'S(z ) < 7r } is not a connected subset of C, S is simply connected. The presence of Coo rather than C is essential in the statement of 7. 1 .21 .

7.1.23 THEOREM. (Mittag-Leffler) FOR AN OPEN SUBSET U OF COO , IF S C U, S· n U = 0, AND FOR EACH a IN S THERE IS A RATIONAL

clef N(a) cn (a) FUNCTION ra (Z) = Ln=1 (Z _ a)n ' FOR SOME f IN M(Q) , P(f) = S AND

Pa (f) = ra · PROOF. For {Kn}nEN as in 4. 10.9, the sets

are finite, and Pn (z) � L ra (z) is a rational function holomorphic in an aESn

open set containing Kn- I . Since Sn is finite, Coo \ Sn has only one component Cn and 00 E Cn. Thus Runge's Theorem (7.1 .11 ) applies and yields a rational function Pn for which 00 is the only pole: Pn is a polynomial.

1 Furthermore, the Pn may be chosen so that I Pn (z) - Pn (z) 1 < 2" on Kn- I , and f � PI + L�=2 (Pn - Pn) meets the requirements. 0

7.1 .24 THEOREM. (Weierstrafi) IF

U E O(q , S c U, AND S· n U = 0 ,

FOR SOME F IN H(U) , S c Z(F) .

PROOF. If the set S = ° resp. S = {anh�n�N<oo ' then F == 1 resp.


meets the requirements. Thus only the possibility 5 = {an} nEN requires attention. The paradigm for F when 5 = {anhSnSN<oo, motivates the argument below.

The components of U are pairwise disjoint regions. If Ie is the solution for the component C of U and for z in C, I(z) �f !c (z) , then I is a solution for U.

Thus, for ease of presentation, it is assumed that U itself is a region and 0 tJ- 5. The situation for which 0 E 5 is dealt with in 7. 1 .26c) . The region fl �f U \ 5 is polygonally connected (1 .7. 11 ) .

In 7. 1.23, when Pn (z) �f __ 1_ and a is fixed in fl, for a nonempty z - an D(z, rt contained in fl, and a polygon 7r; connecting a to z and contained

in fl, the function

In,7rz ; D(z, rt 3 z' r-+ exp (1 + 1 I Pn (Z) dZ) 7rz [z ,z 1

IS m H [D(z, r)O ] . If 1]; is a polygon like 7rz , then 1 - 1 Pr> (Z) dz is in 7T"z 1]z 27riZ, whence In,7rz (z') is independent of the choice of 7rz : In,7rz �f In .

If Fn �f exp (fn ) , then ( z -(an ) ' = 0, and direct calculation shows Fn z) z - a that Fn (z) = ___ n . For appropriate polynomials {Pn} l < < , the series a - an _ 11. CXJ 00

PI + L (Pn - Pn) � I E M(fl) . Furthermore F �f exp(f) is well-defined n=2

in fl, F E H(fl) , and

F(z) = -- . __ n exp - Pn(W) dw . z - al II { z - a [ 1 ] } a - al n2':2 a - an 'Y

(7. 1 . 25)

The right member of (7. 1 . 25) is an infinite product. Owing to the continuity of exp, the infinite product converges uniformly on compact subsets of fl and as z -+ an F (z) -+ O. Hence the set 5 consists of removable singularities of F and if F (an ) �f 0, n E N, then F E H (U) and 5 C Z(F). 0

The discussion of infinite products is given in Section 7.2 where the development implies Z(f) = 5, v. 7.2 .11 .

[ 7.1 .26 Note. a) If I i s required to have a zero of multiplicity f.-ln at an, the sequence {an} nEN may be modified so that each an appears f.-ln times.


b) If f is entire, f(O) i- 0, a = 0, and Z(f) �f {an}nEN ' then S· = 0 and l an l -+ 00. Hence, for each n, if I z l < l an l , then

1 1 (Xl zk Pn = - = - = - '"'" -z - an ( z ) � a�+l . an 1 - - k=O an Kn ( ) k The approximating polynomial Pn is , for some Kn, - L : k=O n

and the convergence inducing factor exp (- 1 Pn(W) dw ) takes

the form exp L z k+l . The exponent L z k+l ( Kn k+l ) Kn k+l k=O (k + l )an k=O (k + l )an

is a partial sum of the familiar pOwer series representation of a de-termination of - In ( __

1_)

. The Mittag-Leffler Theorem comz - an bined with the argument above yields the WeierstrajJ product representation for the entire function f:

f(z) = II { (I - : ) exp (t zk+l k+l) } . n� l n k=O (k + l )an (7.1 .27)

c) If f(O) = 0, for some k in N, the product representation is preceded by a factor zk . Since the right member of (7. 1 .27) converges uniformly on compact subsets of U and since the function exp is continuous, the validity of (7. 1 .27) is automatic. Its derivation is independent of the theory of infinite products.]

7.1.28 Exercise. If [Z (fW = au, there is no function F such that : a) F is holomorphic in an open set V that properly contains U; b) Fl u= f. Thus au is a natural boundary for f.

[Hint: For such an F, i f Z(Ft = 0, then U = C. If Z(Ft i- 0 5.3.52 applies. ]

7.1 .29 Exercise. If U E 0(((:) , then H (U) contains an f for which au is a natural boundary.

[Hint: In U there is a sequence S �f {zn}nEN such that S· = au. In H (U) there is an f such that Z(f) = S.]

Section 7.2. Infinite Products 313

7.1.30 Example. If a sequence of holomorphic functions converges everywhere, need the convergence be uniform on every compact set? In [Dav] the following construction uses Runge's Theorem to produce a sequence {Pn} nEN of polynomials such that

lim Pn(Z) = { 1 n--+oo 0

if Z = 0 otherwise · (7. 1 .31)

Although the sequence converges uniformly on every compact set not containing {O} , the sequence fails to converge uniformly on every compact set properly containing {O} .

For n in N, if Un �f { a + ib : a > -� , I bl < � } while

clef ( ) \ clef [ 1 ] Fn = D O , n Un and Kn = Fnl:.J ; , n ,

each Kn is compact , Kn C K�+l ' and U Kn = C \ {O}. Furthermore, for nEN

each n, C \ Kn is connected and there are disjoint Open neighborhoods Vn of Kn and Wn of o. Hence there is a function gn holomorphic in Vnl:.JWn and such that

if z E Wn if Z E V;, .

Polynomial Runge (7. 1 .14) implies there is a polynomial Pn such that on 1 Kn, Ign - Pn l < - . Hence (7. 1 .31 ) is valid. n

7.2. Infinite Products

The Weierstrafi product representation (7. 1 . 27) leads naturally to a discussion of infinite products . 7.2 . 1 DEFINITION. FOR A SEQUENCE {an}nEN OF COMPLEX NUMBERS ,

00 THE infinite product P �f II ( 1 + an ) EXISTS IFF a) n=l

N 1· II (1 ) def 1 · 1m + an = 1m PN

N --+00 N--+oo n=l def 1 · b) EXISTS, IN WHICH CASE P = 1m PN ; FOR SOME no ,

N--+oo

N lim II ( 1 + an)

N--+oo n=:no


EXISTS AND IS NOT ZERO. [ 7.2 .2 Remark. The condition b) has the following motivations.

• A product of finitely many factors is zero iff at least one factor is zero, whereas , e.g. , if an == -� each factor 1 + an is nonzero

N 1 and yet lim II (1 + an ) = lim -N = 0. N--+= N--+= 2 n= 1 • An infinite series converges iff every subseries arising from the

deletion of finitely many terms converges . The validity of the analogous statement for infinite products is assured by b). If

{-I an = 1

if n = 1 otherwise '

N N then lim II (1 + an) = ° but lim II (1 + an ) does not N--+= N--+= n=1 n=2 = exist . The condition b) eliminates II (1 + an ) from consid-

n= 1 eration as an infinite product .

As the developments below reveal, the simpler definition requiring N only that lim II (1 + an ) exist suffices in the context of repreN--+= rL= l senting entire functions as infinite products.]

= 7.2.3 Example. If the series L bn converges and 1 + an = exp (bn ) , then

n= 1 = lim bn = 0, lim an = 0, and the infinite product II (1 + an ) converges to

rL---.--t cx)

7.2.4 Exercise. If {anL:SnSN <= C C:

n= 1

qn �f g (1 + l an l ) � exp (� I an l) ; IpN - 1 1 � qN - 1 .

[Hint: If x 2: 0, the relation

x2 1 + x � 1 + x + 2T + . . . = exp(x)


applies for the first inequality. Induction leads to the equation

which deals with the second.] ex:>

7.2 .5 THEOREM. IF: a) X IS A SET; b) {an}nEN C eX ; c) L l an (x) 1 CON-n=1

VERGES UNIFORMLY ON X; d) sup l an (x) 1 < 00, THE INFINITE PRODUCT

00 nE/'! x E X

II [1 + an (x)] CONVERGES UNIFORMLY ON X AND DEFINES A FUNCTION n=1 f IN eX FOR ANY PERMUTATION 7r : N 3 n f-t 7r (n) E N,

00 f(x) = II [1 + a7r(n) (x)] . n=1

FURTHERMORE, f(b) = 0 IFF FOR SOME no , 1 + ano (b) = O. [ 7.2.6 Remark. The hypotheses c) and d) are independent . For example, if X = [0, 1] and:

•

if x E (0, 1]

otherwise x and an (x) = 2n ' n 2: 2, c) holds and d) does not .

• If an (x) == 1 , n E N, d) holds and c) does not .]

PROOF. For some M, sup l an (x) 1 :::; M < 00. If K E N, there is an N n E/'! xEX

depending on K and such that ( 1 , 2, . . . , K) C [7r( I ) , 7r (2) , . . . , 7r (N)] . If 1 ex:> - > E > 0 there is a Ko such that sup L lan (x) 1 < E. If 2 xEX n=Ko

K N K 2: Ko , PK �f II [1 + an (x)] , and nN �f II [1 + a7r(n) (x)] , n=1 n=1

owing to 7.2.4,

00 I n N - PK I .:::; IPK I (eC - 1) :::; IPK I L En < 2 1pK I E.

n= 1 (7.2 .7)


Moreover, 7.2.4 implies that for some P and all K, IPK (X) I :::; P < 00. If 7r is the identity permutation, (7.2 .7) implies that for some f, PK � f. Furthermore, if K > Ko , then

IPK - PKo I :::; 2 1pKo I E, IPK I 2': (1 - 2E) IPKo I , {x E X} '* { I f(x) 1 2': ( 1 - 2E) IpKo (x) I } .

Hence {J(x) = o} {} {pKo (x) = a} . Finally (7.2 .7) implies that for each x, lim nN (X) = lim PN (X) . 0 N--+= N--+=

7.2.8 COROLLARY. IF 0 :::; an < 1, n E N, THEN

N = PROOF. If PN �f II ( 1 - an ) , for some p, PN ..l- p. If L an < 00, 7.2.5

n= 1 n=1 = implies P > O. On the other hand, if L an = 00, for each N,

n=1

N whence lim II (1 - an ) = o. N--+= n=1

[ 7.2.9 Remark. The last sentence provides another motivation for b) in 7.2 .1 .]

o

ex:> n If z E 1U the series - L -=- converges, say to l(z) . If g �f exp(l ) , then n n= 1

g'(z) (1 - z)' ( g (Z) ) ' . . ( ) , whence -- = o. Smce l(O) = 0, g (z) = 1 - z, I .e . , g z 1 - z 1 - z

(1 - z) exp[-l(z)] = 1 and so l (z) is a determination of In(l - z) . It follows that for z fixed in 1U and K in N, for some N(K, z) in N,

= (N(K'Z) k ) = � 1 - (1 - z) exp t; � :::; � 2- K < 2.

= (N(K,Z) n) According to 7.2.3, II (1 - z) exp t; z

n converges .


More generally, if {an} nEN C C and 0 < I an I :::; I an+1 1 t oo, for any fixed Z there is a sequence {N(K, n , z ) }KEN such that

00 ( Z ) (N(K,n, z) (:n ) k) II 1 - - exp L an k n=1 k=1

converges to some number J(z) . If N(K, n, z) can be chosen to depend only on n , i.e. , if there is a sequence {Nn} nEN such that

converges , J : C 3 z f-t J(z) is entire and 7.2.8 implies J(b) = 0 iff for some n, b = an.

The expression EN(Z) �f ( 1 - z) exp (� z:) , the product of 1 - z and the exponential function of the sum of the first N terms of the Maclaurin series for a particular determination of - In( 1 - z) , is approximately 1 (1 - z) . -- = 1. The next result serveS to estimate the error of the apl - z proximation.

7.2 .10 LEMMA. IF I z l :::; 1 AND N E Z+ , THEN 1 1 - EN (Z) I :::; I z I N+1 . PROOF. There are nonnegative numbers Ck , k E Z+ such that

Furthermore, En (O) = 1 and there is a sequence {ddkEN of nonnegative

numbers such that E�(z) = _zN (1 + f dkZk) . If k=1

00

-1 E� (u) du

( ) �f 1 - EN (Z) _ [o ,z] g N Z - Z N +1 - --'-'�z'-;N:-;-+-:-1 --,

then gN (Z) = L ekz\ ek 2': 0, k E N. If I z l :::; 1 , then IgN (Z) 1 :::; gN (I) = 1 . k= 1

o


7.2 .11 Exercise. a) If 0 < l art l :::; l an+l l t oo, then N contains a sequence

{Nn}nEN such that for each positive R, � C:, I ) Nn+l < 00. Further-

more,

(7.2.12)

converges for all z in C. b) If, for some k in N, ( R ) k+l

� � � k + 1 n� l < 00 (7.2.13)

for all positive R, then

(7.2.14)

converges for all z in C. c) The infinite products in (7.2.12) resp. (7.2. 14) represent entire func-

tions F resp. G such that Z(F) = Z(G) = {an}nEN (v. 7.2 .5) . [ 7.2.15 Remark. If (7.2 .13) obtains for some k in Z+ , there i s a least such k, say h, the genus of the function G. When k = h , the representation (7.2 . 14) is the canonical product representation for G. More generally, when p is a polynomial function, G and h are as described above, and m E N+ , then eP(z) zTnG(z) is a canonical product .]

7.2.16 Exercise. If m E M(C) there are entire functions f and g such that off P(m) �f {Pn }" EN' m = � .

[Hint: It may be assumed that the poles of m are listed according to their orders , i .e . , if p is a pole of order k, then p appears k times in the listing of P(m). There is an entire function g such that Z(g) = P (m) . ]

7.2 .17 Exercise. If f E E and Z(f) = 0, for some entire function w, f = exp(w) .

[Hint: The functions f ' and w : C 3 z r-+ w(z) �f r f'((t)) dt f J[o.z] f t

are entire and ( exp)w) ) ' = 0.]


7.2.18 Exercise. If 1 E E, for some k in N, and some entire function w , I(z) = zk exp[w(z)] II ENn(a) (�) .

aEZ(f) The preceding discussion shows that if 0 < l an l t oo, although an infi-

nite product 11. (1 - :n) may fail to converge, nevertheless , with the aid

of the convergence inducing factors, exp (t zk+l k+ l ) , n E N, the

k=O (k + l )an

infinite product IT (1 - �) exp (t zk+l k+ l ) does converge. The

n= 1 an k=O (k + l )an next paragraphs motivate and introduce Blaschke products that deal with the problem of finding a function 1 such that 1 E H (1U) and Z (I) IS a preassigned subset of 1U.

The Identity Theorem 5.3.52 implies that if 1 =j=. 0, then

Z(ft n 1U = 0.

If Z(f) is infinite, it is countable, i .e . , Z(f) �f {a" }nEN' and thus l an l -+ 1 , whence the assumption l an l t 1 i s admissible. A natural conjecture for the form of 1 (z) is the infinite product

DC (7 .2 .19)

n= l

If (7 .2 .19) converges , lim (an - z) = 1 . The preceding experience with n--+= infinite products suggests that some convergence inducing factors {Fn } nEN can force the relation

lim (an - z) F,, (z) = 1 n--+oo

and, more to the point , the convergence of

00

II (an - z) Fn (z) . n= 1

7.2.22 Exercise. If 0 < l an l t 1, and

clef l an l ( 1 ) Fn(z) = ( _ ) = sgn (an) _ , an 1 - an z 1 - an z

(7.2 .20) holds .

(7 .2 .20)

(7.2 .21)

(7.2 .23)


7.2.24 THEOREM. FOR Fn AS IN (7.2 .23 ) , IF

00 L (1 - l an l ) < 00 (7.2.25 ) n= l

AND 0 < l an l t 1 , THE BLASCHKE PRODUCT 00 ( 00 ) k k an - Z

Z II (an - z) Fn (z) = Z II sgn (an) _ n=l n= l 1 - anz (7.2.26)

CONVERGES UNIFORMLY ON COMPACT SUBSETS OF 1U (THUS REPRESENTS A FUNCTION B IN H (1U) ) AND Z(B) = {an}nEN l:J{O} .

[ 7.2.27 Remark. Since

(v. the discussion of l (z) in 7.2.9) , the F" have forms reminiscent of the convergence inducing factors encountered in the Weierstrafi Product Theorem. In the latter the arguments of exp are finite sums, whereas the corresponding arguments in the Fn are infinite series.]

clef an - Z PROOF. If (7.2.25) holds , I z i :s; r < 1 , and bn (z) = 1 - sgn (an) _ , 1 - anz then

I bn(z) 1 = I ( 1 - I an l ) (an + l an l z) I an ( 1 - anz) = (1 - lan l ) 1 1 - sgn (a)z I :s; (1 _ l an l ) 1 + r

' 1 - anz 1 - r

whence 7.2.5 applies to the sequence { l bn (z)+ l }nEN " o 7.2.28 Exercise. If E c 1I', then 1U contains a sequence {an} nEN such that EC = ({an} nENr and the Blaschke product (7.2.26 ) converges. (In particular 1I' can be a natural boundary for the function B represented by (7.2.26 ) . )

Section 7.2. Infinite Products

[Hint: When k = 2, 3, . . . and ° :::; m :::;: k - 1, if

Gkrn clef { iO = re k 2m71" 2( m

k+ 1 )71" } ,

1 - T :::; r :::; 1 , -k- :::; () <

if Ee n G krn i- (/) otherwise

for some enumeration {an L"EN of {bkrn }2"Sk<=, o"Srn<k- l , a) l an l :::; l an+l l , n E N; b) Ee = ({an}nENr; c ) (7.2.26) converges.]

321

The following paragraphs offer a development that leads to an elementary proof that (7.2.25) is not only sufficient but is also necessary for the convergence of the Blaschke product.

h . z - an clef ( ) f h f T e argument centers on the negatIves -1

_ - = cPa" Z 0 t e ac-- anz tors appearing in the Blaschke product . These are of independent interest, e g., in the theory of conformal mapping, v. Chapter 8. 7.2.29 Exercise. If l a l < 1: a) cPa E H (1U) ; b)

{ I z I :::; I } '* {cPa 0 cP -a ( z) = z} ; c) cPa (1U) = 1U; d) cPa (1I') = 1I'.

7.2.30 Exercise. If l a l < R, then <t>a,R : D(O, Rr 3 z f-t RcP]'; (�) IS a holomorphic autojection of D(O, Rr .

7.2.31 LEMMA . (Jensen) IF 1 E H (1U) , Z(f) n 1U ::) {an} l"Sn"SN<oo ' AND I clef SUp II(z) = M < 00, THEN

I z l < l

N clef II h i ) . d PROOF. If g = cPa" (z) , t en - E H (1U . For z fixed m 1U an n=l g

(7.2.32)


if f.-l < r < 1 , the Maximum Modulus Theorem implies

Furthermore 7.2.29d) implies that if I z l = 1 , then I g(z) 1 = 1 and

lim min I g(w) 1 = 1 . Ttl Iwl=T

7.2.33 COROLLARY . (Schwarz's Lemma) IF

THEN :

J E H (1U) , sup IJ (z) 1 :::; 1 , AND J(O) = 0, I z l < l

I J (z) l :::; I z l if I z l < 1 ; 1 1' (0) 1 :::; 1 .

o

( 7 . 2 . 34) ( 7 . 2 . 35 )

IF EQUALITY OBTAINS IN ( 7 . 2 . 34) FOR SOME NONZERO z, OR IF EQUALITY OBTAINS IN ( 7 . 2 . 34) , FOR SOME REAL (J, J(z) == eiOz . PROOF. The inequalities follow from 7.2.31 (n = 1 , al = 0, and M = 1) .

J(z) def . If -- = h(z) , then h E H (1U) . The MaXImum Modulus Theorem z implies in the current context that if I z l < 1 and I h (z) 1 = 1, then h is a constant function and I h(z) 1 == 1 .

If 1 1' (0) 1 = 1 , then h(O) = 1 . The Maximum Modulus Theorem applies once more to show that h is a constant function. 0

7.2.36 COROLLARY . IF , IN (7.2.34) , EQUALITY OBTAINS FOR SOME z IN N

1U, FOR SOME (J IN JR., J(z) == MeiO II cPan (z) . n= l

PROOF. Schwarz's Lemma (7.2.33) applies to the function

N

Hence, if I z l < 1 and J(z) = M II cPan (z) , for some 1] in 1I', k(z) = 1]Z. n= l

7.2.37 Exercise. If r > 0, J E H [D(O, rt] ' J(O) i- 0, and

Z(f) n D(O, rt = {al , . . . , an } ,

o

Section 7.3. Entire Functions 323

then (7.2.38)

7.2.39 THEOREM. IF 0 < l an l < 1, n E N, AND THE BLASCHKE PRODUCT 00

II sgn (an) an � CONVERGES (THUS REPRESENTING A FUNCTION 1 n=1 1 - anz 00

NOT IDENTICALLY ZERO AND IN H (1U) ) , THEN L ( I - I an l ) < 00 . n= 1

PROOF. Since 1(0) i- 0, 7.2.31 implies that for each N,

N 0 < 1 1(0) 1 < II l an l n= 1

and 7.2.8 applies.

7.2.40 Exercise. If 10'1 < 1, then 4>� (0) = 1 - 10' 1 2 ; 4>;, (0') = 11 1 2 ' 1 - a

7.3. Entire Functions

o

Since a nonconstant entire function 1 cannot be bounded, a study of the behavior of max I I(z) 1 �f -M'\R; I) , particularly for large R, is in order.

I z l=R R. Nevanlinna [NevI] developed this subject in a very significant manner. Only the introductory aspects of the material are treated below.

7.3.1 LEMMA . AN ENTIRE FUNCTION 1 IS A POLYNOMIAL OF DEGREE NOT EXCEEDING n IFF FOR SOME POSITIVE CONSTANT e, M(R; I) :s; eRn .

PROOF. If I(z) = aozn + . . . + an and ao i- 0, for large R,

M (R; I) < 2 1 a I . Rn - 0

Conversely, if, for some positive constant e, M(R; I) :s; eR" , for large I z l , II (z) 1 :s; 1 + Cl z l n . The Cauchy formula for the coefficients in the power

00 series L Ckzk representing 1 implies that for all large R, if k > n, then

k=O

(7.3.2)


n As R t 00 (7.3.2) implies Ck = 0, whence J(z) = L CkZk . o

k=O . M(R; f) Thus only those J such that for each n m N, sup = 00, merit

R>o Rn

attention. A convenient classification of the orders oj growth of entire functions is achieved as follows .

clef - In ln M(R- f) 7.3.3 DEFINITION. WHEN J E E , p(f) = lim 1 ' IS THE order

R-+= n R clef - In M(R; f) oj J. IF p(f) E (0, 00) , THEN T(f) = lim (1) IS THE type oj p(f) .

R-+= Rp 7.3.4 Exercise. If M(R; f) = exp (RT ) , then p(f) = r. 7.3.5 Exercise. If M(R; f) = exp (tRa ) , then T(f) = t .

An entire function J is representable as a power series, i .e . ,

J(z) = L cnzn , n=O

If clef iO h 1· I convergent for all z in C. z = re , t en 1m I Cn rn = 0, and for some n-+=

if n � vf (r) , if n > vf (r)

,

vf (r) is the central index and f.l,f (r) = ICVf (T) rVf (T) I · Thus CVf (T) ZVf (T) may be regarded as the last maximal term of the series. 7.3.6 Exercise. a) If p > 0, then

M(r ; pf) = pM(r ; f) and f.l,pf (r) = pf.l,f (r) . b) As a function of r, vf (r) is a right-continuous Z+-valued step-function and if J is not constant , vf (r) t oo as r t 00.

7.3.7 LEMMA. a) p(f) = inf { ). : IJ(z) 1 < exp ( I z l ), ) } �f w(f) ; b) WHEN p(f) < 00 AND

K(f) � { /'£ : /'£ > ° and for large r, M(r; f) < exp (/'£rP) } i- 0,

THEN T(f) = inf K(f) �f /'£(f) . In ln M(r; f) PROOF. a) If E > 0, for large r, 1 < p(f) + E, whence n r


The Maximum Modulus Theorem implies that for all z ,

Hence w(f) :::; p(f). On the other hand, if (J" < p(f) , for some {r n} nEN '

rn t oo and M (rn ; f) > exp (r�) , whence (J" < w(f) : w (f) = p(f) .

b ) For large r , InM(r; f) < IWP(f) , whence T(f) :::; /'£(f) . If (J" < /'£(f) , for some {rn } nEN ' rn t oo and M (rn; f) > exp ((J"r�(f) ) , whence (J" < T(f) : T(f) = /'£(f) . 0

A sense of the behavior of p and T is provided by the functions in clef t r 7.3.8 Exercise. a) If r, t > 0 and J(z) = e Z , then

p(f) = r and T(f) = t ; b) If J(z) �f eCz , then p(f) = 00; c) If p E C[z] , i .e . , i f p is a polynomial, then p(p) = 0; d) If p E C[z] and J �f eP , then p(f) = degp; e) If 0 < Iq l < 1

00 and J(z) �f L qn2 zn , then J is not a polynomial although p(f) = O.

n=O 7.3.9 Exercise. a) If a > 0 and g(z) �f J(az) , then p(g) = p(f). b) If 0 < p(f ) , then p[exp(f)] = 00. c) p (f') = p(f ) .

- ln ln /-lj (r) 7.3 .10 THEOREM. IF J E E, THEN p(f) = lim I . T--+oo n r

PROOF. The Gutzmer coefficient estimate implies

((f) �f lim ln ln/-lj (r) :::; p(f). T--+oo In r If ((f) = 00, then p(f) :::; ((f) . If ((f) < (3 < 00, for large r,

I en I r n :::; /-l j ( r) < exp (rf3)

and if r = (�) � and n is large, then l en l rn < ( e(3:f3 ) � . If 2e(3 �f 15, then

l8T� J 00 M(r; f) :::; L len l rn + L len l rn n=O n= r8T� 1

00 < (r5rf3 + 1) /-lj (r) + L T � �f I(r) + II(r)

n= L8T(3J


whence

-. In ln M(r; I) In ln[I(r) + II(r)] hm < , T-HJO In r - In r

lim II(r) = o. T --+ =

If 0 < x :s; y and r is large, the inequality x + y :s; 2y yields

In[I(r) + II(r)] :s; In 2I(r) , In I(r) :s; In (28ri3 ) + In l-lf (r) = In 28 + ,B ln r + In l-lf (r ) .

If some Cm ¥= O and r i s large, 7.3.7b) implies I-l (r) > I cm l rm ; 7.3.7a) implies p(pl) = p(l) and ((pI) = ((I) . Hence it may be assumed that I Cm I > 1 , and so In I-l f (r) > m In r. If m > ,B, the estimate applied earlier yields In 28 + ,B In r + In I-l f (r) :s; 2 1n I-l f (r) and

whence ((I) ? p(l) .

In In I ( r ) In 2 + In In I-l f ( r ) ---'---'- < , In r - In r

clef � ( n ) - ;.; 7.3. 11 THEOREM . IF a > 0 AND Fa (z) = � ae zn , then n=l

p (F,, ) = a and T (Fa ) = 1 .

PROOF. The maximization techniques of the calculus show

o

Because the coefficients in the power series representation of Fa are positive,

= n

M (r; Fa ) = L (:J - � rn , n=l a monotonely increasing function of r, whence

(F ) - 1· In ln M (r; Fa) p ", - 1m . T--+= In r

From 7.3.10 it follows that p (Fa ) = a. Furthermore, if E > 0, for large r, -dn r < In ln M (r; F", ) - a ln r < dn r and

_( In M (r; Fa ) r < < r( . ra o


The behavior of an entire function f is related to its Weierstrafi product representation, in particular by the way in which Z(f) is distributed throughout C.

Here is a list of some entire functions correlated with their orders and the cardinalities of their sets of zerOs:

f sm z eZ eez n

Lakzk k=O

sm z

p(f) 1 1

00

0

1

#[Z(f)] No o o n

The picture that emerges is far from clear because in the list no attention is paid to the relative density of the set of zeros, e .g. ,

#[Z(f) n D(O, R)] R

Two useful measures of the frequency of occurrence of the zeros of f clef { } clef ( ) are, when a = an nEN = Z f ,

clef . { " I } v(a) = mf a : a E JR., � l an l" < 00

un EZ(f)\ {O} clef = the exponent of convergence of a,

8(a) �f sup { m : m E Z+ , L la 11 Tn = 00

} anEZ (f)\{O} n

�f the exponent of divergence of a.

7.3.12 Exercise. If F(z) �f IT Eh (:n ) is a canonical product (of genus

h) , then h :s; v(a) :s; h + 1 and h :S; p(F) .

7.3.13 Exercise. I f p(f) < p(g) , then p(fg) = p(g ) . I f p(f) = p(g ) , then p(fg) = p(f) [= p(g) ] . 7.3.14 Exercise. a) For some sequence a, l an l t oo and v(a) = 8(a) = 00 .

b) The numbers v(a) and 8(a) are both finite or both infinite.

7.3.15 LEMMA . IF f E E AND Z(f) = {an }nEN �f a, THEN v(a) :s; p(f) .


PROOF. If p(f) = 00, the result is automatic. If p(f) < 00, it may be assumed that l an l :::; l an+l l and that J(O) = 1 (whence l al l > 0) .

Since #[Z(f)] = No, J is not a polynomial function. Thus, for each n in N, if r is large, then M (r; f) is large. (For a polynomial function J, rn #[Z(f)] is finite and neither v (a) nor J(a) is defined. If the definition of v (a) is extended in a natural way to apply to finite sequences , for any finite sequence a, v (a) = -00 < 0 = p(f) , i.e . , the result is automatic.)

If n is large, l a� l n < 1 . Thus, for large n, large r, and positive E, I � I < M(r; f) <

exp [rp(f)+< j. a� - rn rn

exp [rp(f)+< j ( e[p(f) + E] ) P(f)+' The minimum of is , achieved when rn n

which is large, since n is large,

Hence I a1n I < (e [p(f� + E] ) P(t\+, , and if J > E, then

00 1 1 I P(f)+,s L - < 00. an n=l

Furthermore, E and J are arbitrary positive numbers . The conclusions above are encapsulated in

7.3.16 THEOREM. (Hadamard) IF

o

FOR SOME POLYNOMIAL P SUCH THAT deg(p) :::; p(f) , SOME k IN Z+ , AND SOME canonical product P(z) (cf. 7.2. 15) ,

J(z) = exp[p(z)]zk P(z) .

PROOF. Since p(f) �f P < 00, v (a) + J(a) < 00. In ( z ) clef ( z ) E,s(a) an

= 1 - an exp [Pn (z) ] ,


Pn is a polynomial and deg (Pn ) = J(a) . Then 7.2.10 implies

whence IT E8(a) (:n ) converges and defines an entire function F. For

I(z) . . some ), in N, ZA F( z) i- ° and hence, for some entIre functIOn p,

n(R) 1 (z) = ZA g (1 - :n )

x [exp [P(z) + � P, (z)] JL, (1 - :. ) cPO " '] �f PR(Z) x qR(Z) .

In D(O, R)O , In (1 - :n ) may be determined so that

n(R) ex:> and if hR(Z) � p(z) + � Pn (z) +

n=r�) +l [In (1 - �J + Pn (Z)] , then

qR(Z) = ehR (z) . If

Z E D(O, Rt and n > # [Z(f) n D(O, 2Rrl �f n(R) ,

then I �, I < � . The power series representing In ( 1 - :n ) implies

I ( Z ) I 2R8(a)+1 In 1 - - + P n ( z) :::; 8 ( ) 1 · an l an l a + Hence the definition of J (a) implies that for large n in the infinite series representing hR, the terms are dominated in absolute value by the terms

00 2R8(a)+1 of '""' ) . � l an l 8 (a +1 If R > 1 and I z i = 2R, then IpR(z) 1 2': 1 , whence

M(2R; f) 2': M (2R; qR) .


Thus in D(O, R) , � [hR(Z)] < In M(2R; I) , and according to 5.3.34, if

00 hR(Z) �f L brr/n ,

m=O

In M(2R; I) - � (bo) . . . then I bm I < 2 , m E N. DIrect calculatIOn shows that If Rrn

p(n) (0) LCX) 1 n > p, then bn = --,- - - and n. nan k=n(R)+l k

I p(n) (o) 1 < 2 In M(2R; 1) - � (bo) + � _1_ , - Rn � I I n. (7.3 . 17)

n. k=n(R)+l an

If p + E < n, then In M(2R; I) < (2R)P+€ and since n > v(a) , as R -+ 00,

both terms in the right member of (7.3 .17) approach zero. It follows that if n > p, p(nl (O) = 0, i .e . , p is a polynomial deg(p) :::; [pl . 0

7.3.18 COROLLARY . IF f E E, p(f) < 00, AND p(f) tJ. N+ , i .e . , IF

p(f) E [0, (0) \ N+ ,

FOR EACH a IN C, #[Z(f - a)] 2': No · ( f - a ) PROOF. If #[Z(f - a)] < No , for some polynomial g , Z -g � = (/) and 7.3.16 implies for some polynomial p, f - a = eP • Hence

p(f - a) [= p(f)] = p.

[ 7.3.19 Remark. In 7.3.18 there are resonances with the Weierstrafi-Casorati Theorem (5.4.3c) ) and the Little Picard Theorem (v. Chapter 9) and the general phenomenon of defective functions . ]

o

The development above is rounded out by the following items. The first is an application of contour integration. The others are formulre that relate in a direct way the density of the zeros of an entire function f to p(f) .

7.3.20 Example. The existence and evaluation of

Section 7.3. Entire Functions

turn on the existence , for E in (0, 1 ) , of the (improper) integral,

For small positive t,

l In 1 1 - eit I I = � l In 1 2 - 2 cos t i l

� � (In 2 + 2 1 ln I sin � I I ) � l�

2 + l In I � I I ' and 1E I ln xl dx = - lim (x ln x - x) I �= E - dn E. Hence J exists.

° 8to

331

For z fixed, z - w as a function of w is in H (1U) and is never Zero in 1U. Hence In I z - w i E Ha (1U) = MVP(1U) , and so

1 1271" - In I z - eit l dt = In I z i . 271" 0

Furthermore, In Iw - 1 1 and In Iw - zl are negative, but Iw - 1 1 < Iw - z l , whence l In Iw - 1 1 1 > l In Iw - z l l ·

In Figure 7.3.1 , 0 < r < 0.5, z = 1 + r, and w lies on the arc ABC.

0 - l + r = z

Figure 7.3 .1 .


In '][' \ {AlB}, l In Iw - zl l is bounded. In sum, for w in '][' and z in [1 , 1 .5] ' l In Iw - zl l is dominated by an integrable function. Hence a passage to the limit as r ..l- 0 (and z -+ 1) is justified. Because lim In I z l = 0, it follows that

1 1271" . 1 1271" . z--+ l

lim - ln l z - e,t l dt = - In I 1 - e,t l dt = 0. rto 27r 0 27r 0 7.3.21 THEOREM. (Jensen's formula) IF

f E H (D(O, Rn , 0 < r < R, f(O) i- 0, AND (f) n D(O, r) = {a l , " " aN } (A ZERO OF ORDER k OCCURS k TIMES) , THEN

N 1 1271" In I f (O) 1 + 2:)n -I r I = - In I f (reiO) I dO.

n= l an 27r 0 [ 7.3.22 Remark. Jensen's formula is an analog of the PoissonJensen formula (6 .2. 1 1 ) .]

PROOF. There are three cases. The first provides the context for the other two.

Case 1 . If f i- 0 in D( a, Rt , Jensen's formula is valid in the circumstances because In I f I E HaIR [D( a, Rt] ' and thus Jensen's formula is an expression of the MVP.

Case 2. If Z(f) n D(O, r) c D(O, rt , for any function g for which Z(f) n D(O, r) = Z(g) n D(O, r) , Case 1 applies to L �f h: g

In l h(O) 1 = 2� (1271" In I f (reiO) I dO) - 2� (1271" ln lg (eiO) 1 dO) . (7.3.23)

The choice of g is rather unrestricted. Owing to the special and useful N

properties of the functions cPa (v. 7.2.29) , g (z) �f II [-cP�n (z)] may be n=l

used. Then (7.3.23) is

1 (1271" ) 1 (1271" N

) In l h(0) 1 = 27r 0 In l f (reiO) l dO - 27r 0 �ln lcParn (eiO) l dO . (7.3.24)

Since I cP"rn (eiO ) I = 1 , at most the first term in the right member of (7.3.24) is not Zero.

Section 7.3. Entire Functions

Furthermore,

N In I h(O) 1 = In 11 (0) 1 - 2:)n I <p� (0) I ,

n=l

N In I h(O) 1 = In 11 (0) 1 + � In

IL l ·

333

Case 3. If Z(f) n 8[D(a, r)l i- 0, Z(f) may be enumerated so that

1 0' 1 1 :::; · · · :::; l am l < r and l am+l l = . . . = l aN I = r. r In Jensen's formula, In

l an l = 0, m + 1 :::; n :::; N. The g in Case 2 is re-

placed by ?i defined by the formula

- clef I . and h by h = :::; . For some s m (r, R) , g

h E H [D(O, stl and Z (h) n D(O, st = 0. Thus

H ·f clef iO < < N h owever, 1 an = re n , m _ n _ , t en N

In Ih (reiO) I = In I I (reiO) 1 - L In 1 1 - ei (O-On ) I · n=m+l

From 7.3.20 it follows that for each n ,

1271" o In 1 1 - ei (O-On ) I dO = O.

Consequently

(7.3.25)

(7.3.27)

(7.3.28)

(7.3.29)


The contents of (7.3.26)-(7.3.29) imply the required conclusion. 0 [ 7.3.30 Note. Jensen's formula is a valuable tool in the extended study of E, v. [NevI] , and of the Hardy spaces

HP (lJ) , 0 � p � 00,

as defined below, v . [Hil , Rud] .

When f E H (lJ) , and

In + t �f { In t if t 2: 1 o if t < l '

exp {2� 127r

In+ If (reiO) I dO} if p = 0 1 { 2

� 127r If (reiO) I P dO} P if 0 < P < 00 '

sUPO-S097r If (reiO) I if p = 00

then Mp (f; r) as a function of r increases monotonically on [0, 1 ) , v. [Ge3, PS , Rud] . Moreover,

HP (lJ) �f { f : f E H (lJ) , I l f l lp �f lim Mp(f; r) < 00 } . ] r ..... l

7.3.31 Exercise. If n(r; I) �f #[Z(f) n D(O, r) ] and f(O) = 1: a) n(r; I) � In M(2r; I) ;

b) lim In n(r; 1) � p(f) . r ..... = In r

[Hint: Jensen's formula applies.]


7.4. 1 Exercise. If lJe C fl, f E M (fl) , and f (1I') c 1I', then f is a rational function. 7.4.2 Exercise. If f E M (([= ) , f is a rational function.

= = 7.4.3 Exercise. If L I an - bn l < 00, then II : = �: converges in

71= 1 n=l


and represents a function in M(Q) . = = n

7.4.4 Exercise. For I(z) �f � zn and, when a > 0, g(z) �f � (:!)O! ' what are vf (r) and vg (r)?

7.4.5 Exercise. a) p(f) = inf { a : a 2: 0, lim I I (z) l l , z ,=r.:s: e 1 z 1a } . b) r ..... = = If I(z) �f LCnzn, by abuse of notation when Cn = 0,

n=O - n ln n

p(f) = lim _ 1 I I . n---+CXJ n en 1 - £ill. c) If p(f) < 00, then T(f) = -(I) lim n I cn l n •

ep n ..... = 7.4.6 Exercise. (The Open Mapping Theorem for meromorphic functions) If 1 is meromorphic in a region Q, then I (Q) is an open subset of C= .

= 7.4.7 Exercise. If I(z) = LCnzn and S(f) n 1Uc = S(f) n 1I' = P(f) ,

then sup I cn l < 00. nEN n=O

[Hint: # (S n 1UC ) < 00.]

7.4.8 Exercise. If 1 E [H (1U)] n [C (1UC , C) ] and I I(z) l l lI'= K, 1 is a rational function.

[Hint: The Schwarz Reflection Principle and the Cauchy-Riemann equations apply.]

7.4.9 Exercise. a) For the meromorphic function

z - z T : C \ {-i} 3 z f-t --. , z + z T (n+) = 1U. b) Is there an entire function 1 such that 1 (n+) = 1U?

8 Conformal Mapping

8.1. Riemann's Mapping Theorem

In each of 5.5.8-5.5. 11 , 5.5.17-5.5.19, 7.1 .21 , and 7.1.22 a simply connected region Q plays a central role.

Combined with 8 .1 .1 below, the contents of the cited results provide a useful edifice of logically equivalent characterizations (v. 8 .1 .8) of simply connected regions in C.

8.1 .1 THEOREM. (Riemann) IF Q IS SIMPLY CONNECTED AND Q i- C, FOR SOME UNIVALENT f IN H(Q) , f(Q) = 1U.

[ 8 .1 .2 Remark. The result 8 .1 .1 was stated by Riemann. It was first proved by Koebe who created an algorithm for constructing a sequence {fn}nEN of univalent functions in H (Q) . He showed that for some univalent f in H (Q) , fn � f on each compact subset of Q and f(Q) = 1U.

Riemann's Mapping Theorem is frequently called the Conformal Mapping Theorem. The term conformal refers to the fact that the mapping f preserves angles (v. 8. 1.6, 8 .1 .7) .

The PROOF below consists of the crucial 8 . 1 .3 LEMMA followed by the main argument . The line of proof is nonconstructive (existential) and is based on the Arzela-Ascoli theme as expressed by Vitali's Theorem (5.3.60) . Other tools in the argument are Schwarz's Lemma (7.2.33) and the functions cPa used in the study of Blaschke products (v. 7.2 .29-7.2.31 and 7.2.36) .]

PROOF .

8.1 .3 LEMMA. IF Q IS SIMPLY CONNECTED AND Q i- C, FOR SOME UNIVALENT g IN H (Q) , g(Q) c 1U.

336

Section 8.1 . Riemann's Mapping Theorem 337

PROOF of 8 .1 .3. If (C \ flt i- 0, even if fl is not simply connected, for some b and some positive r, D(b, rt c C \ fl and 9 : fl 3 z f-t �b meets

z -the requirements.

On the other hand,

fl �f C \ (- 00 , 0] = { z : z = ReiO , R > 0, -Jr < (J < Jr }

is also simply connected but (C \ flt = 0. The function 9 above is useless. For this fl and in general, the importance of simple connectedness is revealed.

If a tJ. fl, for some h in H (fl) , [h(zW = z - a (v. 5.5. 19) . If

then Z1 - a = Z2 - a, whence Z1 = Z2 : h is univalent . The Open Mapping Theorem (5.3.39) implies h(fl) contains some a

other than 0, hence, for some small r in (0, 1 0' 1 ) , 0 tJ. D(a, r) Co h (fl) . If Z2 E fl and h (Z2 ) E D( -a, r ) , then -h (Z2 ) E D(a, r) and so for some Z1 in fl, h (zd = -h (Z2 ) . But, as shown above, Z1 = Z2 ,

h (zd = -h (zd = 0 E D(a, rt,

a contradiction: D(-a, rt � h (fl) . Hence, if z E fl, then I h(z) + 0'1 > r and clef r 9 = h + a meets the requirements. 0 PROOF of 8 .1 .1 . Vitali's Theorem (5.3.60) implies that the nonempty family F of univalent maps of fl into V is precompact in the II l lao-induced topology of Hb (fl) , the set of functions bounded and holomorphic on fl.

For b fixed in fl, if k E F, then h �f � [k - k(b) ] E F and h(b) = O. If

h'(b) = I h' (b) 1 eiO and 9 �f e-ioh, then 9 E F and g'(b) > O. Hence attention is focused on the non empty set 9 of functions 9 such that:

9 is univalent in fl, g(fl) c V, g(b) = 0, and g'(b) > O. If M = sup { g' (b) : 9 E 9 }, for some sequence {gn} nEN contained in

9, g� (b) t M. Hence, for some subsequence, again denoted {gn}nEN' and some 9 in Hb(fl) , gn � 9 on each compact subset of fl and g� (b) t M. Furthermore (v. 5.3.35) , lim g� (b) = g' (b) = M < 00 . n ..... =

The next arguments show that: a) 9 is univalent , whence 9 E 9; b) g(fl) = V.

a) . Since each gn is univalent , M > O. If Zo E fl, for each n,

338 Chapter 8. Conformal Mapping

Hurwitz's Theorem (5.4.39) implies

g(z) - g (zo ) == ° or [g (z) - g (zo )] l n\{zo } i- 0.

If g(z) == g (zo ) , then g' (b ) = 0 < M, a contradiction. Hence, for any Zo in Q, [g(z) - g (zo) ] l n\{zo } i- 0: g i s univalent .

b) If l e i < 1 and e tJ. g(Q) , the simple connectedness of Q enters the

argument again: 5.5.19 implies G : Q 3 z f-t J cPc (g (z) = g(z) - e is 1 - cg (z)

well-defined and in H (Q) . Direct calculation shows G is univalent on Q and G(Q) C 1U.

G(z) - G(b) Finally, if H : Q 3 z f-t cPO(b) (Z) = , then H E 9 . Since 1 - G(b)G(z)

Ii-:::I 2 Ii-:::I , G'(b) 1 + l e i (1 - y l e i ) = 1 - 2y l e i + l e i > 0, H (b) = I (b) 1 2

= fi:I ' M > M, 1 - G 2y l e i a contradiction. 0

8.1 .4 DEFINITION. WHEN (J E [0 , 27r) , THE straight line through a at inclination (J IS clef { °O } L( a, (J) = z : z = a + tet , t E ffi. .

8.1 .5 Exercise. If L is a straight line in C and a E L, for a unique (J in [0 , 27r) , L = L (a, (J) .

8 .1 .6 Exercise. (Conformality, first version) If

f E H (Q) , a E Q, J' (a) i- 0, and L �f L (a, (J)

: a) For t in [0, 1] , the equation /'(t) = f (a + teiO) defines a curVe through f(a) . b) For the line L (f(a) , cP) , tangent to /'* at f(a) ,

cP - (J E Arg [J' (a)] .

c) For two differentiable curve-images intersecting at a (whence their fimages intersect at f (a) ) , the size of the angle between their tangents at a and the size of the angle between the tangents to their f -images at f (a) are the same.

[Hint: b) The chain rule for derivatives applies to the calculation of /,' .]

Section S.l . Riemann's Mapping Theorem

8 .1 .7 Exercise. (Conformality, second version) If

f �f u + iv E H (Q) , J'(a) i- 0,

and "/ �f x + iy is a differentiable curve such that ,,/(0) = a: a) x'(O)el + y'(0)e2 is a vector parallel to the tangent line at ,,/(0) ;

clef clef b) for U = u 0 "/ and V = v 0 ,,/ ,

U' (O) = ux (a)x' (O) + uy (a)y' (O) , V'(O) = vx (a)x' (O) + vy (a)y'(O) ;

339

c) U'(O)el + V'(0)e2 is a vector parallel to the tangent line at f 0 ,,/(0) ;

d) for some ¢ in [0, 27r) , the matrix ( ux((a

)) uy

((a

)) ) is a multiple of the Vx a Vy a

(orthogonal) matrix ( co� ¢A-Slll � ) ;

- Slll ,/-, cos ,/-, e) the vector U'(O)el + V'(0)e2 is the vector x' (O)el + y'(0)e2 rotated

through an angle of size ¢.

[Hint: d) The Cauchy-Riemann equations apply.] The two versions of conformality may be reworded as follows. If f E H (Q) and f is invertible at a, f preserves angles at a and the sense of rotation at a. Central to the phenomenon are the Cauchy-Riemann equations, i.e . , the differentiability of f. The material in 10.2.46 is related to the current discussion.

8.1.8 THEOREM. FOR A REGION Q, THE FOLLOWING STATEMENTS ARE LOGICALLY EQUIVALENT: a) Q AND 1U ARE HOMEOMORPHIC; b) Q IS SIMPLY CONNECTED ; c) IF a E Coo \ Q AND "/ IS A LOOP FOR WHICH "/* C Q, Ind -y (a) = 0; d) Coo \ Q IS CONNECTED ; e) IF f E H (Q) , FOR SOME SEQUENCE {Pn}nEN OF POLYNOMIAL FUNC

TIONS, Pn � f ON EACH COMPACT SUBSET OF Q; f) IF f E H (Q) , "/ IS A RECTIFIABLE LOOP , AND "/* C Q, THEN

i f dz = 0,

i.e. , IF "/ IS A RECTIFIABLE CURVE AND "/* C Q, THEN i f dz DE

PENDS ONLY ON ,,/(0) AND ,,/(1 ) AND NOT ON THE PARTICULAR CURVE "/;


g) IF 1 E H (Q) AND 0 tic I(Q) , FOR SOME F IN H (Q) , F' = I; h) IF 1 E H (Q) A N D 0 tic I(Q) , FOR SOME G IN H (Q) , 1 = exp(G) ; i) IF 1 E H (Q) , 0 tic I(Q) , AND n E N, FOR SOME H IN H (Q) ,

I = Hn .

[ 8.1 .9 Remark. Listed below are implications already established and their provenances. These implications and their derivations are the root of the subsequent argument. The entire set is intimately related to 8 .1 . 1 . ]

Implication b) '* d) b) '* e) b) '* f) b) '* g) b) '* h) b) '* i)

Provenance 7.1.21 7.1.19

5.5. 16c) 5.5.17 5.5.18 5.5.19

PROOF . a) '* b) : If "( is a loop, "(* C 1U, and J(t) == 0, then clef F(t, s ) = S"((t)

is a homotopy such that "( '" F,1U J . If \11 : 1U r-+ Q is a homeomorphism and r is a loop such that r* c Q, then \11- 1 0 r �f "( is a loop such that "(* C 1U and \11 0 F 0 \11- 1 �f <t> is a homotopy. Furthermore, if Ll(t) == \11(0) , then r "'''' ,n Ll.

b) '* c) : v. 5.5. 16c) . c) '* d): v. the PROOF of 7. 1 .21 . d) '* e) : v. 7.1. 14. e) '* f ) : Every polynomial is a derivative and f) obtains for derivatives. f) '* g): If {a, z} C Q, there is a polygon n connecting a to z and a

corresponding "( such that "(* = n . For the map

F : Q '3 z r-+ F(z) �f 11 Ib(t)] d"((t) ,

F'(z) = I(z) . Owing to f ) , F(z) is well-defined, Le. , is independent of the choice of "( so long as "((0) = a.

f' clef f' g) '* h): For some F, F' = 7 ' If <t> = exp(F) , then <t>' = <t>7 ' whence

(<t» ' <t> 7 = 0 and so 7 is a nonzero constant, say C. For some c, C = exp( c) ( clef and 1 = exp <t> - c) = exp(G) .

Section 8.1 . Riemann's Mapping Theorem 341

h) '* i): H = exp ( �) . i) '* a) : If Q -j. C, since the heart of the PROOF of 8 .1 .1 is the existence

z of H when n = 2, a) follows. If Q = C, the map C '3 z r-+ --1-1

E 1U is a 1 + z

homeomorphism. D

8.1 .10 THEOREM. IF f IS A conformal self-map (holomorphic autojection) OF 1U, FOR A (J IN [O, 27r) AND AN a IN 1U, f = eiO¢a (WHENCE f- 1 E H (1U) , i.e. , f IS biholomorphic) .

PROOF . If f is a holomorphic autojection of 1U, for some (unique! ) b in 1U, f(b) = 0.5. Hence g �f ¢0.5 0 f is a holomorphic autojection of 1U, and g(0.5) = o. Since ¢-b 0 g(b) = b, Schwarz 's Lemma (7.2.33) applies. D

8.1 .11 Exercise. For f as in 8.1 . 10, what is f- 1 ?

8.1 .12 Exercise. If Q is a simply connected proper subregion of C and a E Q there is no unique conformal map f of Q onto 1U and such that f(a) = o. There is a unique conformal map f such that f(a) = 0 and f'(a) > o.

[Hint: If f and g are two maps as in 8 .1 .10 applies to f o g- I . ]

8.1 .13 Exercise. In 8 .1 .12 , the f such that f(a) = 0 and f'(a) > 0 is the unique solution of the problem of finding in F an f such that

f' (a) = sup { h' (a) : h E F } .

In effect, 8 .1 .13 restates the Riemann Mapping Theorem as a result in the calculus of variations . Lemma 8.1 .3 provides a motivation for Riemann's original attempt to prove 8 .1 .1 . According to 4.7.18, if f : Q r-+ 1U is univalent and holomorphic, the Cauchy-Riemann equations imply that the area of f(Q) is

A[J(Q)] �f in 1 f' (z) 1 2 dx dy

= in (u; + u�) dx dy = in (v; + v�) dx dy ::::: 1 .

Hence, if A[J(Q)] i s maximal, e.g., i f A[J(Q)] = I , f i s a good candidate for the biholomorphic map of Q on 1U. Euler 's equations for the stated variational problem take the form t:m = l1v = 0 and the boundary conditions for u and v are simply that for all a in 8(Q) , lim lu (z) 1 V lim I v (z) 1 ::::: 1. The

z=a z=a discussion in 6.2.17-6.2.22 implies that if Q is a Dirichlet region, for some u, l1u l n= 0, u l a(n)= lim lu(z) I , i.e. , Dirichlet's problem has a solution.

z=a


Riemann's proof of 8 .1 .1 involved the implicit assumption that there is a solution to the problem of maximizing In (u; + u�) dx dy subject to

the normalizing condition, In (u; + u�) dx dy � 1. The assertion that this kind of variational problem has a solution became known as Dirichlet 's Principle .

However there appeared 8.1 .14 Example. (Weierstrafi) Among all 1 in COO(lR, lR) such that I(x) = 0 if I x l ::=: 1 and 1' (-0.9) = -1'(0, 9) = 1 there is none for which

11 {t2 + [J' (t) ] 2} � dt is least. (The problem is to minimize the length of a curve "( : [0 , 1] '3 t r-+ t + il(t) such that "((0) = -"((I) = -1 , "( t o. The geometry of the situation shows that the infimum of all such lengths is 2 but that the length of each such curve exceeds 2.)

Thus Dirichlet's Principle, as a statement about the solvability of a variational problem, was suspect.

In the hands of (alphabetically) Hilbert, Koebe, Konig, Neumann, Poincare, Schwarz, Weyl, and Zaremba, the validity of Dirichlet's Principle for a simply connected fl achieved a semblance of validity. In modified form, Dirichlet's Principle is central to one of the derivations of the Uniformization Theorem, v. 10.3.20.

Given the validity of Dirichlet's Principle for a si�ply connected fl, Riemann's approach leads to a u and a harmonic conjugate v such that 1 �f u + iv is the required conformal map of fl onto 1U. The intimate connection of Dirichlet's Principle to the solvability of Dirichlet's problem seems to bind the two to questions about Dirichlet regions, barriers, simple connectedness, etc. The PROOF of 8 .1 .1 resolves these questions.

The discussion in Section 8.5 is also germane to the considerations above. For the harmonic function g corresponding to a Green's function G(· , a ) , v. 8.5.1-8.5.9, some harmonic conjugate, say h, of g leads to a function ¢ �f h + ig E H (fl) . If, for z in fl, I(z) �f (z - a) exp[¢(z)] , then 1 is a conformal map of fl onto 1U and I(a) = o.

8.2. Mobius Transformations

If Z; , 1 :::; i :::; 4, are four elements of C, the number

is their cross ratio or anharmonic ratio. For Z2 , Z3 , Z4 fixed and pairwise different , X (z, Z2 , Z3 , Z4 ) �f M(z) is a function on Coo \ {Z3 } and may be

Section 8.2. Mobius Transformations 343

extended to Coo by defining M (Z3 ) to be 00 . There are constants a, b, c, d az + b clef such that M(z) = cz + d = Tabed(Z) and

More generally, when l1 -j. 0 , the map

az + b Tabed : C '3 z r-+ ---d cz +

is a Mobius transformation. By definition,

Tabed ( -�) = 00 and Tabed(oo) = � . Correspondingly, for e as in Section 5.6, there is

� clef - 1 2 { } 2 Tabed = e T"b(.d8 : L: \ (0, 0 , 1 ) r-+ L: ,

which may be extended by continuity to a self-map of L:2 • When ambiguity is unlikely, the subscript abed is dropped. Note that if a -j. 0, then

1 whence, if a = V75. ' then (a a) (ad) - (ab) (ac) = 1 , and as the need arises the value of l1 may be taken to be 1 .

8.2.1 Exercise. Each Mobius transformation T is invertible and

- 1 -dz + b Tabed : C '3 z r-+ = T(-d)be(-a) . cz - a

(Thus each T is one-one: {T(z) = T (z') ) {} {z = z'} . ) 8.2.2 Exercise. a) The set of all Mobius transformations Tabed is a group M with respect to composition 0 as a binary operation.

b) Those for which l1 = ad - bc = 1 is a normal subgroup M1 containing the normal subsubgroup E �f {TW01 ' T(- l )OO(- l ) } .

c ) The quotient group MdE, denoted Mo, is isomorphic to SL(2, q , the multiplicative group of all 2 x 2 matrices M with entries from C and for which det (M) = 1 .


zP

Iw - Z l2 = I Z - a l · I ZP - Z l = I Z - a l · I ZP - a l - I Z - a l 2

= I Z - a l · I ZP - a l - (r2 - IW - ZI 2 ) , I Z - a l · I ZP - a l = r2 .

Figure 8.2 .1 .

clef r2 . When r > 0, z -j. a, and k = 1 1 2 ' the pomt z - a

clef zP = a + k(z - a)

is the reflection or inversion of z in Ca(r) �f a [D( a, r rl and z is the reflection of zP in Ca(r) : z = (zP)P . By abuse of notation, aP = 00 and ooP = a. Figure 8.2.1 above illustrates the geometry of reflection or inversion in the circle Ca(r) .

For a line L( a, 0) the reflection of z � a + 1 z - a l eiq, in L( a, 0) is

P clef 1 1 -i(q,-20) Z = a + z - a e

and z is the reflection of zP in L( a, 0) .

8.2.3 Exercise. The reflection of z in the line L(O, O) IS z . For L(a, O) regarded as a mirror, zP is the mirror image of z.


The superscript P serves as a generic notation for a reflection z r-+ zP performed with respect to some circle or line.

8.2.4 THEOREM. IF z E C AND ad - be -j. 0, THEN Tabcd (Z) ARISES FROM THE PERFORMANCE OF AN EVEN NUMBER OF REFLECTIONS (IN LINES OR CIRCLES) .

PROOF. The argument can be followed by reference to Figure 8.2.2 . For any z , TOl lO (z) arises by reflecting z in '][' and reflecting the result in JR.

If b -j. 0 there are (infinitely many) pairs of parallel lines L1 , L2 , separated by I�I and perpendicular to the line through b and o. Direct calculation reveals that T1b01 (Z) arises by reflecting z in L1 and reflecting the result in L2 •

o

Z1: the reflection of z in L1 z12: the reflection of z1 in L2

Figure 8.2.2.


If ° :::; (J < 27r, then TeiOOOl (z) arises by reflecting z in L3 and reflecting the result in L4 •

If ° < A E lR, then TA001 (Z) arises by reflecting z in '][' [= Co (I) ] and reflecting the result in Co (VA) . Finally,

{ r::. + be - ad if e -j. ° T. (z) - e e( ez + d) abed - a b

z + if e = O d d D

8.2.5 Exercise. The elements (z, z') in C� are a pair of mutual reflections in a circle C resp. a line L iff X (z', ZI , Z2 , Z3 ) = X (z, ZI , z2 , Z3 ) is meaningful, , i.e. , # (z' , ZI , Z2 , Z3 ) = # (z , ZI , Z2 , Z3 ) = 4, and true.

8.2.6 Exercise. The validity of X (z', ZI , Z2 , Z3 ) = X (z , ZI , Z2 , Z3 ) is independent of the choice of an acceptable triple ZI , Z2 , Z3 .

8.2.7 Exercise. If T E M \ {id }, the number of fixed points of T is 0, 1 , or 2.

1 8.2.8 Exercise. If ° < 10' 1 < 1 the fixed points of 4;" are ± -- . sgn a 8.2.9 Exercise. If {ZI ' Z2 , Z3 } resp. {W I , W2 , W3 } are two sets of three points in Coo , for precisely one T in Mo, T (Zi ) = Wi , 1 :::; i :::; 3.

8.2.10 Exercise. (Extended Schwarz Reflection Principle) If: a) r > 0; dcl { e } b) A = a + ret : 0 :::; (Jl < (J < (J2 :::; 27r ; c)

1 E H [D(a, rt] n C [D(O, atl:.JA, q ;

d) 1 (A) C Cb (R) ; e) z r-+ zPa resp. z r-+ ZPb is the map z r-+ zP performed with respect to Ca(r) resp. Cb (R) ; f)

{ I(z) F(z) �f [J (zPa Wb

I(z)

if I z - a l < r if I zPa - a l < r , if z E A;

then Q �f D(a, rtl:.JAl:.J [D(a, rtJPa is a region and F E H (Q) .

8.2 .11 Exercise. If K is a circle lying on L2, then 8 (K \ { (O, O, I ) } ) is a circle or a straight line. 8.2.12 Exercise. The group M is generated by the subset

1 To : C \ {a} '3 z r-+ - , Tab : C '3 z r-+ az + b, a, b E C. z


8.2.13 Exercise. a) If .c is the set of all circles and straight lines and T E M, then T(.c) = .c. b) If D is the set of all open discs and the complements of all closed discs and T E M, then T (D) = D. 8.2.14 Exercise. If a, b, c, d E lR and ad - bc > 0, then Tabed leaves n+ invariant: Tabed (n+) = n+ .

8.2.15 DEFINITION. FOR A REGION Q, Aut (Q) IS THE SET OF CONFORMAL AUTOJECTIONS OF Q. 8.2.16 Exercise. If Q is a region, with respect to composition 0 as a binary operation, Aut (Q) is a group. 8.2 .17 Example. According to 8 .1 . 10,

Aut (V) = { T : T(z) = eiO¢,, (z) , a E V, 0 -::; (J < 27r } , a proper subgroup of M. 8.2.18 Exercise. a) I E Aut (C) iff for some constants a and b, I(z) == az + b.

b) I E Aut (Coo ) iff I E M. c) If I E Aut (V) , then 11(0) 1 = 1 1- 1 (0) 1 .

8.2.19 Exercise. For Z1 and Z2 in a region Q and a subgroup G of Aut (Q), the relation rvG defined by {Z1 rvG Z2 } {} {Z1 E G (Z2) } is an equivalence relation.

The set QIG (the quotient space ) consists of the rvG-equivalence classes of Q. 8.2.20 Exercise. The set VI Aut (V) is a single point. If k E N and wk = 1 , then G �f { wn : n E N } i s a finite subgroup that may be identified with a subgroup of Aut (V) . The set VIG may be identified with the open sector

{ reiO : 0 -::; r < 1 , 0 < (J < 2: } .

The operator ( III ) ' 1 ( III ) 2 Sch : H (Q) '3 I r-+ 7' - "2 7' �f {f, }

is the Schwarzian derivative. The operator

) ' ) ' clef Lg : H(Q '3 I r-+ (ln l = Lf

is the logoid derivative. If I' (z ) -j. 0, for any determination of In [I' (z ) ] ' I" 7' = {In [I' (z) ] }' is unambiguously defined. Hence both {f, z} and L f (z) are well-defined if I' (z ) -j. o.


8.2.21 Exercise. a) {J 0 g , z} = {J, g} [g' (z)] 2 + {g, z} ; b) for a Mobius transformation T, {T, z} = 0 and {T 0 g, z} = {g, z} ; c) Laf+b = Lf;

Items b) and c) above serve as motivations for introducing Sch and Lg : Sch is Mobius-invariant while Lg is invariant with respect to an important subgroup of M. The next result is the converse of b) above.

8.2.22 LEMMA. IF {J, z} = 0, THEN f E M. PROOF. If F � Lg (I) , because {J, z} = 0, i t follows that 2F' = F2. Hence

1 2 '

i.e., for some constant a, Lg (I)(z) = F(z) = (ln !')' = _ _ 2_ . Hence, for z - a any determination of In(z - a) , [-2 In(z - a)l ' = (In !') ' . Successive inte

A grations imply that for constants A and B, f = -- + B. D z - a There follows an interesting link between simple connectedness and M.

8.2.23 THEOREM. IF Q IS SIMPLY CONNECTED SUBREGION OF COO AND Aut (Q) c M, FOR SOME T IN M, T(Q) IS C, Coo OR, 1U. PROOF. Since Coo with one point removed is equivalent to C, via some T in M, only the possibility that # [800 (Q)] 2': 2 needs treatment. If 00 E Q, for some T in M, T(Q) C C (v. 8.2.9) . Hence the assumption Q c C is admissible.

There is a biholomorphic bijection h : 1U r-+ Q such that h' (0) = 1 . If h E M, then h(1U) (= Q) is an open disc or an open half-plane, i.e. , Q is conformally equivalent to an open disc. The next argument shows that h E M.

If g E Aut (1U) and f �f h o g 0 h- 1 , then f E Aut (Q) c M, f o h = h o g, and {J o h, z} = {h o g, z} .

Moreover 8.2.21a) and 8.2.21b) imply

{h , g} [g'(z) ] 2 = {h , z} . (8.2.24)

If w E 1U, g = ¢-w and z = 0 , then (8.2.24) reduces to

{h , w} (1 - lw I2 ) = {h , O} . (8.2.25)

iO Hence, if {h , O} � ReiO , for w in 1U, {h, w} =

1 �lwI2 ' i.e. , by abuse of

notation, {h, 1U} c L(O, O ) . Off Z (h' ) , {h , w} is holomorphic in w. The

Section 8.3. Bergman's Kernel Functions 349

Open Mapping Theorem implies that the map 1U '3 w r-+ {h , w} is a constant map: {h , w} == C. Thus, for all w in 1U, C (1 - lw I2 ) = {h , O} . As Iw l t 1 , {h , O} = C -+ O. By virtue of 8.2.22, h E M. D

8.3. Bergman's Kernel Functions

The existence of a conformal map 1 of a simply connected proper subregion fl of C onto 1U leads, via Bergman's kernels described below, to an explicit formula for the mapping function I. The starting point for the development is the study, for any region fl, of SJ(fl) �f L2 (fl, ),2 ) n H (fl) . As a subspace of L2 (fl, ),2) , SJ(fl) is naturally endowed with an inner product ( , ) and an associated norm I I 1 1 2 , the latter providing a metric.

8.3.1 Exercise. If 1 �f u + iv E H (fl) : a)

b) for the map F : ]R2 '3 (x, y) r-+ (u, v) E ]R2 , the derivative F' (v. Section 4.7) exists and p (F') = u;, + u� = 1 1' 1 2 .

The next result, despite its negative character, reveals something useful about SJ(fl) .

8.3.2 LEMMA. IF fl IS A REGION AND a E fl, THEN

SJ(fl) = SJ(fl \ {a}) �f SJ (fla ) .

PROOF. If 1 E SJ(fl) , then 1 E H (fla ) and 1 1 1 1 2 d),2 = r 1 1 1 2 d),2 then n ina SJ(fl) C SJ (fla ) .

1 On the other hand, if 1 E H (fla ) \ H (fl) , e.g. , if I (z) = -- , for some z - a positive R, A(a; 0, Rr C fla and I IA(a;O ,R) O is represented by a Laurent series: 00

(8.3.3) n=-(X)

Just as in 5.3.28, if z - a �f reiO , 0 :::; (J < 27r, and 0 :::; r < s < min{ l , R}, then II (z) 1 2 = L Cncrnrn+rnei(n-rn)O and

n,rnEZ

(8.3.4)


The only nonzero inner integrals in the right member of (8.3.4) are those for which n - m = o. Hence

(8.3.5)

For at least one term of the series in the right member of (8.3.3) , n < 0 since otherwise 1 E H (Q) . The corresponding integral in (8.3.5) is divergent and thus r II (zW d)..2 = 00: 1 tic SJ (Qa ) . D ina

8.3.6 COROLLARY . IF 0 -j. 1 E E , THEN [ 1 1 1 2 d)..2 = 00 : SJ (C) = {O} . PROOF. If 0 -j. I, then 1 i s represented by a power series

00 I (z ) = L cnzn

n=O

convergent in C. If R > 0, as in the preceding argument,

For some positive n, Cn -j. o. D

8.3.7 LEMMA. IF Q IS A SIMPLY CONNECTED PROPER SUBREGION OF C, dimSJ(Q) = No .

PROOF. For any bounded region Ql , SJ (QI ) contains the set of all polynomial functions. Thus dimSJ (QI ) = No. If g : Q r-+ 1U is a biholomorphic bijection and 1 E SJ (1U) , then l o g E H (Q) . Furthermore, for the transformation g : Q r-+ 1U, the Cauchy-Riemann equations, 4.7.22, and 4.7.23 imply: a) Idet [J(g) l l = 19' 1 2 ; b) if 1 E SJ (1U) , then l o g · 1 g' 1 E SJ(Q) ; c) if Un } nEN is an orthogonal set in SJ (1U) , then Un 0 g . Ig' I } nEN is an orthogonal set in SJ(Q) . d) {z r-+ zn }nEN is an orthogonal set in H(1U) . D

The next result is the basis for many of the arguments that follow.

8.3.8 LEMMA . IF

1 E SJ(Q) , a E Q, AND r5 �f inf { I z - a l z E 8(Q) } ,

THEN II (a) 1 :::; ��.

Section 8.3. Bergman's Kernel Functions

PROOF.

8.3.9 LEMMA. IF Q IS A REGION, THEN SJ(Q) IS I I 1 1 2-COMPLETE.

PROOF. If # [8(Q)] :::; 1, 8.3.2 and 8.3.6 imply SJ(Q) = {O} .

351

If #[8(Q)] > 1 and {fn }nEN is a I I 1 1 2-Cauchy sequence in SJ(Q) , since L2 (Q, ),2 ) is complete, for some F in L2 (Q, ),2 ) , lim I lln - F I 1 2 = o. n--+ CXJ

If a E Q, for some positive J,

a E 58 �f Q \ [ U D(Z, Jtj , 58 E Sp , and 58 ¥- 0. zE8(fl)

• • 2 7r 2 Moreover, 8.3.8 ImplIes I l fm - In l 1 2 ? -2 Ilrn (a) - In (a) 1 . Hence on Q, m

lim In (z) �f I(z) exists, and on each 58 , In (z) � I(z) . Thus I E H(Q) n--+CXJ and on Q, 1 == F: I E SJ (Q) n L2 (Q, ),2) and lim I l ln - 1 1 1 2 = o. D n---+=

8.3.10 LEMMA. IF {¢" }nEN IS A CON IN SJ (Q) AND I E SJ(O) , THEN

=

n=1 CONVERGES TO I THROUGHOUT Q AND UNIFORMLY ON EACH 58 . PROOF. From 8.3.8 it follows that if z E 58 , then

(8.3. 1 1 )

8.3.12 THEOREM. IF z E 58 AND {¢n }nEN IS A CON IN H (Q) , THEN ex:> 1 L l ¢n (z) 1 2 :::; 7rJ2 n=1

PROOF. If ¢n (z) = 0, n E N, no further argument is needed. That possibilN

ity aside, for some N, the map eN '3 a � (al , . . . , aN ) r-+ L an¢n (z) is a n=1


continuous open map of the Banach space eN onto C. Hence, for some a, N L an q)n (z) = 1 , and A �f { f : f E span (q)1 , . . . , q) N ) , f (z) = 1 } -j. (/). If n=l N N N f E A, then f = L (I, q)n) q)n �f L cnq)n and so 1 = L cnq)n (z) . n=l n=l n=l

Schwarz's inequality in the current context takes the form

N N 1 � L I cn l 2 • L l q)n (z) 1 2 •

n=l n= l

(8.3.13)

Since (8.3.13) is valid for all large N, direct calculation yields the conclusion. D

8.3.14 COROLLARY. THE SERIES (8 .3 . 1 1 ) CONVERGES ABSOLUTELY IN Q.

PROOF. If a E Q there are sequences {an}nEN ' {;3n}nEN of complex numbers such that:

• lan l = 1;3,, 1 = 1, n E N; • {anq)n}nEN is a CON in SJ (Q) ; • anq),, (a) = l q)n (a) l , n E N; • (;3nf, anq)n ) = 1 (1, q)n ) l ·

The Fischer-Riesz Theorem (3.7.14) implies that for some g in SJ (Q) , 00

g = L (;3nf, anq)n) anq)n and dn �f (g, anq)n ) = 1 (I, q)n ) l , n E N. n=l S,h=,,', inequality implie, (� I (t, ¢,, ) I ' 1 ¢,, (a) I) ' <: 1�1�l . D

8.3.15 Exercise. The series 00

(8.3.16) n=l


converges throughout Q2 and if r5 is small and positive, uniformly throughout sl.

[Hint: Schwarz's inequality applies. ] The series (8.3. 16) defines a function

00 K : Q2 '3 (z, w) r-+ L cPn (z)cPn (W) ,

n= l Bergman's kernel. As it stands , K appears to depend on the choice of the CON {cPn} nEW The developments below reveal that K depends only on Q.

8.3. 17 Exercise. IK(z, w) 1 2 :::; K(z, z)K(w, w) . 8.3.18 THEOREM. IF J E S)(Q) AND z E Q, THEN

J(z) = In K(z, w)J(w) dA2 (w ) .

[ 8.3.19 Remark. Because of the preceding equation, K i s a reproducing kernel . ]

PROOF. For z fixed, K(z, · ) E S)(Q) , whence the integral above exists. FurN

thermore, S N denoting L (I, cPn) cPn , n=l

and 8.3 .17 implies

[ 8.3.20 Remark. More generally, if (X, 5 , f.-l) is a measure space, S) is a closed (Hilbert) subspace of L2 (X, f.-l) and for each x in X, the evaluation map 1]x : S) '3 J r-+ J(x) i s in S)' (v. 8.3.8) , Riesz 's Theorem (3.6. 1) implies there is in S) a function Kx such that 1]x (l) = (I, Kx) . The function

2 ) dcl K : X '3 (x, y) r-+ (Ky , Kx = K(x, y)

is a reproducing kernel, i.e. , J(x) = [J( . ) , K( . , x) ] .] In 8.3.21-8.3.23 the underlying context is that of 8.3.20.


8.3.21 Exercise. The reproducing kernel K is unique, i.e. , if K is such that for all f in SJ, f(x) = [J( . ) , K( . , x)] , then K = K. 8.3.22 Example. If SJ = SJ(1U) , then f E SJ iff for the power series repre-

00 00 00

n=O n=O n=O 00 (I, g) = L cndn . The reproducing kernel K in this case is Szego 's kernel:

n=O 1 K(z , w ) = --_- . 1 - zw

[ 8.3.23 Note. From 8.3.18 it follows that for each SJ(fl) there is a reproducing kernel K; 8.3.21 implies that K is unique. Hence K is independent of the choice of the K-generating CON and depends only on fl: K = Kn . ]

8.3.24 Exercise. If fl is simply connected, a E fl, and # [8(fl)] > 1 , for some f in SJ(fl) , f(a) -j. o.

[Hint: For some b in fl and some g in SJ(fl) , g (b) = 1 . For some h in Aut (fl) , h (b) = a .]

IF ( ) �f K ( z, w ) 8.3.25 THEOREM. FOR w IN fl, K(w , w ) -j. O. g z , w - K (w , w ) '

clef { THEN g E B = f f E SJ(fl) , f(w ) = I } AND

I I g l 1 2 = min { l l f l 1 2 : f E B } .

FURTHERMORE, g IS THE ONLY ELEMENT IN B FOR WHICH THE MINIMUM IS ACHIEVED.

[ 8.3.26 Remark. The independence of K from the choice of the CON by which it is generated is reaffirmed by 8.3.25.]

PROOF . If f E B, for some sequence {cn}nEN' 00

n=l 00

n=l


Since w is fixed, (g, cPn ) = :t��2) and I l g l l � = K(�, w) k clef If E B and k - g = h, then h (w , w) = ° and, since h E SJ(Q) , for

some sequence {dn }nEJ'i' 00 00 00

n=l n=l n=l However, k = g + h , whence

8.3.27 Exercise. The last paragraph in the PROOF of 8.3.25 is valid.

D

8.3.28 Exercise. The sequence { '1/Jn : D(O, Rr '3 z r-+ E zn:l } is V -; R nEJ'i

a CON for SJ (D(O, Rr) . 00

[Hint: The map I : D(O, Rr '3 z �f L cnzn is in SJ [D(O, Rr] d ) 7r 2n ] an (f, '1/Jn = -R an- I · n

n=O

8.3.29 LEMMA. IF I : U r-+ V IS A BIHOLOMORPHIC (HENCE CONFORMAL ) MAP BETWEEN TWO OPEN SUBSETS OF C AND SJ (U) -j. {O} , THEN SJ(V) -j. {O} AND: a) BOTH

F : SJ(V) '3 g r-+ 1' . (g o f) AND G : SJ (U) '3 h r-+ (I- I ) ' . (h o 1-1 ) ARE UNITARY , (v. 3.6.16) ; b) EACH OF F AND G IS THE ADJOINT OF THE OTHER, i.e. , [F(g) , h] == [g , G(h)] . PROOF. a) The formula in 4.7.18 for changing variables implies that if g E SJ(V) , then I g l 2 E LI (V, ),2 ) and Ig o 112 1 1' 1 2 E Ll (U, ),2 ) . Hence

I' . (g 0 f) E SJ(U) . Direct calculations lead to the remaining conclusions . D


8.3.30 Exercise. For F and G as in 8.3.29: a)

{h E S:J(U) } '* {F o G(h) = h} , {g E S:J(V) } '* {G 0 F(g) = g} .

b) F [S:J (V)] = S:J(U) and G [S:J(U)] = S:J(V) . c) If <1> is a CON in S:J(U) , then G (<1» is a CON in S:J(V) .

. ( -1 ) ' 1 [Hmt; a) If I(z) = w, then 1 (w) = I'(z). b) The result a)

applies. c) The result 8.3.29a) applies .]

8.3.31 LEMMA. IF 1( · , w) IS A CONFORMAL MAP OF A REGION Q ONTO D(O, R)O AND I(w, w) = 0, !'(w , w) = 1 : a) !' E S:J(Q) ; b)

{¢n (. , w) : Q '3 z r-+ (i [J(z';2]n- l

f'(Z, w) } V :; nEN IS A CON FOR S:J(Q) . PROOF. a) The formula for change of variables (v. 4.7.18) shows

b) The results in 8.3.29 and 8.3.30 apply. D

8.3.32 THEOREM. (Bergman) IF Q IS A SIMPLY CONNECTED PROPER SUB-clef 1 REGION OF C, w IS A FIXED ELEMENT OF Q, AND R = , THE

V7rK(w, w)

lz K(s , w) ds clef FORMULA <1>(z, w) = "-,,,w-

K-(-w-,-w-)- DEFINES A CONFORMAL MAP OF Q

ONTO D(O, Rt . FURTHERMORE, <1>( w, w) = ° and aa<1>

I _ = 1 . z z-w PROOF. Riemann's Mapping Theorem (8. 1 .1 ) implies that there is a conformal map 1 : Q r-+ D(O, R)O , and 8 .1 .12 implies that there is one and only

. al one such I, say I(· , w) for whIch I(w , w) = 0, -(z, w) 1 _ = 1 . Moreover, az z-w 8.3.7 implies that dimS:J(Q) = No . Since K is independent of the choice of a CON for S:J(Q) , the CON {¢n } nEN of 8.3.31 may be used to define K. For that choice,

1 '1/Jl (W) = J7rR, '1/Jn (W) = 0, n > 1 ,

1 al 1 K(z, w) = -2 -a (z , w ) , K(w ,w ) = R2 ' 7rR z 7r <1>(z, w) = I(z, w) . D

Section S.4. Groups and Holomorphy

. 11z ( ) d I VK(w , w) 8.3.33 ExercIse. w K s, w s :::; 7r .

[ 8.3.34 Note. By virtue of 8.3.2, if Q is a region and a E Q, Bergman's kernel functions Kn for Q and Kna for Qa �f Q \ {a} can be identical. If Q is simply connected, Kn provides a conformal map of Q onto 1U, but Kna does not (nor can it) do the same for Qa . Hence, absent the existence of a conformal map f : Q r-+ 1U, the kernel Kn does not serve as the basis for the existence of su�h an f.

On the other hand, if dim SJ (Q) = No, the formula

00 clef ,", --K(z, W) = � ¢n(Z)¢n (W) n= l

is meaningful. However, in the current context , for w fixed,

<1> : Q '3 Z r-+ LZ K(s, w) ds

is not necessarily in H (Q) . Nevertheless, for a maximal simply connected subregion (cf. 5.9.5) Q1 of Q, if w E Ql , as a function of z, <1>(z, w) E H (Qt ) . An extensive treatment of the material in this Section can be found in [Berg] . The applications of Bergman's kernel functions are not confined to the subject of conformal mapping.]

8.4. Groups and Holomorphy

357

The appearance of M, e.g. , via ¢c , in the treatment of 8 .1 .1 , in 8 .1 .10, and in 8.2. 18b) , where the isomorphism of Aut ((Coo ) and M is implicit, suggests that M, regarded as a group with respect to the binary operation 0 : M2 '3 (Tl ' T2 ) r-+ Tl 0 T2 E M, deserves examination. Among the subgroups of M are the following:

W2 . a) when {Wl .W2 } c (C \ {O}) and - IS not real, the set WI

b) the modular group: Mod �f { Tabed : {a , b, c, d} c z} n Mo.


8.4. 1 DEFINITION . A SUBGROUP G OF Mo IS properly discontinuous IFF FOR SOME P IN C, SOME OPEN NEIGHBORHOOD N(p) OF {p} , AND EACH T IN G \ {id } , T(p) tic N(p) .

8.4.2 Exercise. The groups GW1 ,W2 and Mod are properly discontinuous and GW1 ,W2 is a normal subgroup of Mod.

[Hint: If T E (GW1 ,W2 \ {id } ) , p = 0, and

N(p) �f { z : Iz i < min { Iwl ± w2 1 } } ,

then T(p) tic N(p) .

If Tabed E Mod, Tabed -j. id , p = 2i , and

N(p) = { z : Iz - pi < � } , then T(p) tic N(p). ]

8.4.3 Exercise. A properly discontinuous subgroup G of Mo is finite or countable.

[Hint: If #(G) > No , then #[G(p)] > No, p E C and G(pt -j. 0. If q E G(pt , then p E G(qt .]

8.4.4 Exercise. a) When G is a properly discontinuous subgroup of Mo and p and N(p) are the objects in 8.4. 1 , for some nonempty subset 5 of Mo, if S E 5, then S(p) = 00. b) If S E 5 the set

r �f SGS- 1 �f { STS-1 : T E G } �f {T} is a properly discontinuous subgroup of Mo and is isomorphic to G. c) If Tabed E (r \ {id } ) , then Tabcd( (0) -j. 00 and c -j. O.

[Hint: c) . If 00 = T(oo) , then p E [Coo \ N(p)] .] Below, each properly discontinuous group, however denoted-r or G-is assumed to conform to c) in 8.4.4.

8.4.5 LEMMA. IF p AND N(p) ARE THE OBJECTS IN 8.4. 1 , FOR SOME POSITIVE p, IF Iz l > p AND T E (r \ {id } ) , THEN T( (0 ) -j. z.

PROOF. If Tn E r \ {id } , Tn(oo) �f Zn , and I Zn l > n, for some Rn in G,

Tn = SRnS-1 , Zn = SRnS-1 (00) = SRn (p) ,

S- 1 (zn) = Rn (P) E [Coo \ N(p)] , lim S- 1 (zn ) = S- I (oo) = P E [Coo \ N(pW = Coo \ N(p) , n---+oo


a contradiction. 1 8.4.6 Exercise. If Tabed E Mo, then T�bed (z) = .

(cz + dF

359

D

8.4.7 DEFINITION. FOR A GIVEN Tabed IN Mo, WHEN c -j. 0 THE SET { z : I cz + dl = I } IS THE isometric circle Cabed OF Tabed :

THE (CLOSED ) DISC D ( - � , I�I ) IS THE associate OF Tabed . WHEN r IS A PROPERLY DISCONTINUOUS SUBGROUP OF Mo,

clef { d } AND C = - � : Tabed E r .

8.4.8 Exercise. a) The isometric circle of T;;;'�d is the image under Tabed of Cabed. b) If

Tabed E (r \ {id }) and z E Coo , then Tabed(Z) arises by inversion of z in Cabed , a reflection in L, the per-

d a pendicular bisector of the line joining - - and - , and a (possibly trivial) c c rotation centered at � .

c [Hint: For a) , if z E Cabed , for some 8,

-d + eiO a _ e-iO z = and Tabed(Z) = ---c c

For b) , 8.2.4 applies .]

8.4.9 Exercise. a) 00 E Rr ; b) If T E r \ {id } the radii of the isometric circles of T and T- 1 are equal.

8.4. 10 Exercise. If {S, T} c r, ST-1 -j. id , {CS , CT , CST, CS�l , CT� l } is the set of centers, and {rs , rT , rsT , rs� l , rT� l } is the set of radii of the isometric circles of {S, T, ST, S- 1 , T-1 } : a)

rSrT rSTrT r2 rST = l eST - cT I = -- = T . I CT� l - cs l ' rs I CT� l - cs l '

b) l es l < p; c) rs < 2p.


The union of all the discs that are associates of elements of r is contained in a disc of finite radius, say D(O, A) . If I z l > A, then z E Rr .

[Hint: b) : If I cs l > p , then 5-1 (00) = cs ; v. 8.4.5.]

8.4. 11 THEOREM. a) THE UNION U �f U T (Rr ) �f r (R r ) IS DENSE IN TEr

C. b) IF u E Rr AND T E r \ {id } , THEN T(u) tic Rr . c) IF u E 8 (R r ) , N(u) I S A NEIGHBORHOOD OF u, AND v E N(u ) , FOR SOME W I N Rr AND SOME T IN r \ {id } , T(w) = v .

PROOF. a) If U is not dense in C, for some positive r and some u,

D(u, rt C Coo \ u.

Furthermore, for each T in r, T [D(u, rn C Coo \ U. In particular,

T [D(u, rt] c C \ Rr .

Since 00 E Rr , the center o f each isometric circle is in r (R r ) , whence u is not the center of any isometric circle. On the other hand, since u tic R r ,

u is in some D ( - � , I�I) , the associate of some Tabed . Furthermore, 8.4.8

implies that Tabed arises by a reflection in Cabed followed by a reflection in a line and a possibly trivial rotation (a pair of reflections in lines) .

If z E Cabed and the radius of Tabed [D(z, r)] is R , direct calculation r shows R = 1 1 2 2 ' 1 - c r 1 ( d 1 ) °

Since r < � < 2p, if z E D - � , � , then

r clef R > ----,2,.- = kr > r. r 1 - -4p2

Hence the radius Rm of T;bed[D(z , r)] exceeds kTnr, m E N. If m is large, T;bed[D(z, r)] meets Rr , a contradiction.

b) If T E r \ {id } , since u is not in the associate of T, 8.4.8 implies T(u) is in the associate of T-1 , hence is not in Rr .

c) By definition, N (u) meets some Coo \ D ( _ � , I�I ) 0 , i.e. ,

( d 1 ) ° N(u) n D - � ' � = 0.


If D (- � �) is the associate of Tabed, 8.4.8 implies e ' le l

8.4. 12 LEMMA. IF S AND T ARE TWO ELEMENTS OF r , THEN

T (Rr ) n S (Rr ) = 0.

PROOF. Since S- IT E r \ {id } , 8.4. 11b) implies S- IT (R r) n Rr = 0.

361

D

D

8.4.13 DEFINITION . FOR A PROPERLY DISCONTINUOUS GROUP r, A FUNCTION f IN M (UO ) IS r-automorphie IFF FOR EACH T IN r, f 0 T = f.

8.4.14 THEOREM. (Poincare) a) IF Rl IS A RATIONAL FUNCTION,

P (Rd n (et = 0, AND N '3 m > 2,

THEN ON EVERY COMPACT SET DISJOINT FROM P( R) u e· ,

CONVERGES UNIFORMLY AND DEFINES A FUNCTION (h IN

H {Coo \ [P (RI ) u e·l } .

b) IF Tabed E r, THEN (h [Tabed(Z)] = (ez + d)2rn (h (z) . c) IF (h CORRESPONDS TO A RATIONAL R2 , RESTRICTED LIKE R1 , AND, FOR Z NOT IN

clef (h (z) Z (t'h) u P (Rd u e· , F (z) = -(

-) , THEN F IS r-AUTOMORPHIC. (h z

PROOF. a) Since the set e of the centers of the discs associated to the elements of r is bounded, and since the set of radii of those discs is also bounded, there is a positive number p such that for each Sa(3'Yii in R r ,

The circle a [D ( -� , p) ] �f Cabed is concentric with Ca(3'Yii '


All the associated discs are contained in each D ( -� , p ) , whence

the complement of each D ( -� , p) is contained in R r .

If Tabed E Rr , then Tabed (Cabed) results from an inversion in Cabed followed by reflections in lines. The reflections in lines are isometric maps , i .e. , they do not alter distances between points. The inversions in circles multiply distances between points by a constant dependent on I cl (v. PROOF of 8.4.11 ) . Thus the radius of Tabed (Cabed) is pl�1 2 '

The complement of Cabed consists of a bounded open disc and an unbounded component. The unbounded component is mapped by Tubed onto the bounded open disc determined by Tabed (Cubed) . The bounded open disc is a subset of Tabed (Rr ) . Hence, owing to 8.4. 12 , if SOi(3'Yii and Tabed are two maps in Mo, the interior of the intersection of the bounded open disc determined by S"(3'Yii (COi(3'Yii ) and that determined by Tabed (Cabed) is empty. It follows that

(8.4. 15) .

If K is a compact set disjoint from P(R) u e· , for some constant M,

The statements b) and c) follow by direct calculation. R 8.4.16 Exercise. If _1 is not a constant, then F is not a constant. R2

D

When G is a properly discontinuous group, unrestricted by the con-ditions in 8.4.4c) , a fundamental set <t>G in C is a set for which <t>� i- (/) and the statements 8.4.11b)-c) (with r and Rr replaced by G and <t>G) obtain. In particular, Rr is a fundamental set for r .

[ 8 .4.17 Note. In GW1 ,W2 or Mod, there are elements T different from id and for which T' = 1 . For such T there is no corresponding isometric circle and the associate of T is meaningless.]

8.4.18 Exercise. If G is an arbitrary properly discontinuous group and, in the notations introduced above, r = SGS- 1 , then S- 1 R r serves as a fundamental set <t>G for G.

8.4.19 Exercise. a) If G = GW1 ,W2 ' the interior of the parallelogram P determined by O, W1 , W2 together with [0, wI ) U [0, W2 ) is a fundamental

Section 8.5. Conformal Mapping and Green's Functions 363

set <t>G . b) If G = Mod, A �f { z : -� � �(z) < � , 'S(z) > 0 } , B is

the union of the complements of the interiors of all meaningful associates clef { iO 7r 27r } of elements of G, and C = z : Z = e ' 2" � (J � 3 , the union of

D �f (A n B) U C and its reflection DP in lR is a fundamental set <t>G . When G is unrestricted, there are the following options .

• One can work with r �f SGS- 1 for some appropriate S and apply the machinery developed above .

• One can stick with G and cope with complications that can arise when the set of centers of meaningful isometric circles has a cluster point at 00 .

Sometimes, e.g., when G = GW1 ,W2 ' there are no isometric circles, and the resulting discussion is straightforward. The emerging theory is that of elliptic functions to which SOme of the most important contributions came from Weierstrafi [Hil] , v. 8.6.1 1 .

For some properly discontinuous groups r, detailed elaboration of the reasoning behind 8.4.14 leads to the construction [Ford] of r-automorphic functions enjoying special properties, e .g . , having in R r exactly one pole of order one and one zero of order one. Manipulation of such functions leads to others that are holomorphic in n+ and map n+ onto C \ {O, I } . Such functions can be used to prove the Little Picard Theorem (9.3. 1 ) [Rud] .

8.5 . Conformal Mapping and Green's Functions

As noted 6.5.1 if h E H (Q) and 0 tic h (Q) , then In I h l E HaJR (Q) . When Q is simply connected and f : Q r-+ 1U is a conformal map, for some a in Q, f(a) = 0, whence G(· , a) �f - In I f I is not defined at a but is defined in Q \ {a} . 8.5.1 LEMMA . IN THE CONTEXT OF Q, a, AND f ABOVE : a)

G( · , a) E HaJR (Q \ {a}) ; b) IF b E 8oo(Q) , lim G(z, a) = 0; c ) FOR SOME POSITIVE r ,

z---+b

g : Q '3 z r-+ G( z, a) + In I z - a l IS IN HaJR [D(a, rtl . PROOF. a) The Hint in 6.5.1 applies.


b) If fl '3 Z -+ b E 8oo(fl) and In I f(z) 1 -1+ 0, via passage to subsequences as needed, for some 15 in (0, 1 ) , and some {zn }nEN ' fl '3 Zn -+ b while f (zn ) converges to some d in 1U and If (zn) 1 � 1 - 15. For the sequence

of compact sets, Km C K�+1 and U Km = 1U. Owing to the biholomormEN

phic nature of f, each f- 1 (Km) is compact and

Furthermore, 1 .3.7 implies that for some {wn }nEN , Wn -+ C � f-l (d) and f (wn) == f (zn ) . For some large m and some large n, Wn E f- 1 (Km) and Zn E fl \ f- 1 (Km ) , whence f is not bijective. z - a c) The function fl '3 Z r-+ f(z) is in H (fl) . D

8.5.2 DEFINITION. FOR A REGION fl AND AN a IN fl, A FUNCTION G(· , a) CONFORMING TO a)-c) IN 8.5.1 IS (A) GREEN'S FUNCTION FOR fl.

Riemann's Mapping Theorem (8. 1 .1 ) and 8.5.1 imply that for a simply connected proper subregion fl of C and a point a in fl, (a) Green's function G(· , a) exists. The following items delimit to some degree the kinds of regions for which there are and are not Green's functions.

8.5.3 Exercise. If fl is a bounded Dirichlet region and a E fl, there is a Green's function G(· , a) for fl.

[Hint: The solution of the Dirichlet problem for the boundary condition u l a(n/z) = In Iz - a l serves .]

8.5.4 Exercise. If fl = 1U \ {o} � 10 there is no Green's function G(· , O) for fl.

[Hint: The discussion in 6.3. 11 applies.]

8.5.5 Exercise. If G(· , a) is (a) Green's function for fl, then G > 0. [Hint: The Maximum Principle applies.]

8.5.6 LEMMA . THERE IS NO GREEN'S FUNCTION FOR C.

PROOF. If G is a Green's function for C, E > 0 , r > I Z2 - zl l > 15 > 0, and

{ I z - zl l < r5} ::::} { I G(z) - G (zI ) 1 < E} ,


clef G (ZI ) + E . . . then gr (z) = (ln J - ln r) (ln l z - zl l - ln r) IS harmomc m C \ {zd· Fur-

thermore, if A �f A (zl ; J, rt , on 8(A) �f r �f CZ1 (J) u CZ1 (r ) , Gl r:S: gr l r . The Maximum Principle implies GI A:S: gr I A · As r t 00 there emerges

i.e. , G (ZI ) :s: G (Z2 ) : G is a constant and cannot be a Green's functi�n. D

8.5.7 Exercise. a) If G( · , a) is a Green's function for Q, then G( · , a) is unique, and if O(z, a) + In I z - al is harmonic near a, then 0(. , a) > G.

[Hint: The Maximum Principle applies.]

8.5.8 Exercise. If f : Q1 r-+ Q2 is a biholomorphic bijection and G2 ( · , a) is the Green's function for Q2 , then G2( · , a) o f is the Green's function G1 [. , J-l (a)] for Ql .

[Hint: The Open Mapping Theorem applies.]


8.6.1 Exercise. If T E Mo and p, q, r, s are four complex numbers, then

X[T(p) , T(q) , T(r) , T(s)] = X(p, q, r, s ) .

8.6.2 Bxercise. a) If 5 E Mo, and 5-1 {O, 00, 1 } = {Z2 , Z3 , Z4 } , then

b) Four points p, q, r, s are cocircular or collinear iff X (p, q, r, s ) is real.

8 6 3 E . Th · ll· ·ff P - q . 1 . . xerClse. ree pomts p, q, r are co mear 1 -- IS rea .

q - r 8.6.4 Exercise. If Q is a region, 5 c Q, and 5· n Q = 0: a) Q \ 5 is a region; b) SJ(Q) = SJ (Q \ 5) .

[Hint: For b) the argument for 8.3.2 applies.]

8.6.5 Exercise. If Q is a region, S,B '3 5 c Q, Q \ 5 IS a region, and )'2 (5) = 0, then SJ(Q) = SJ (Q \ 5) .

[ 8.6.6 Note. I f 5 = CO! (some Cantor set contained in [0, 1 ] ) , ).2 (5 ) = ° and 1U \ 5 i s a region.

If 5 �f (- 1 , 1 ) , ).2 (5) = ° but 1U \ 5 is not a region.


If 5 �f { z : z = p + iq, {p, q} C QI } , >'2 (5) = 0, but 1U \ 5 contains no region.]

8.6.7 Exercise. If Q � { x + iy : I x l < 1, I y l < 1 } what is the corresponding Bergman kernel K?

[Hint: The Gram-Schmidt algorithm applies to the sequence

8.6.8 Exercise. If Q1 �f { z : 0 :::; a < '25(z) < b :::; 27r } and

clef { iO } Q2 = z : z = Re , 0 < R, a < (J < (3 ,

for some real c, h : Q1 '3 z r-+ eicz is a conformal map of Q1 onto Q2 . 8.6.9 Exercise. If the Schwarzian derivative,

w z (= 2w'(z)w"'(z) - 3 [W"(Z)] 2 ) { , }

2 [w'(z)] 2 '

is regarded as a function of w and z , and T E Mo, then

{w, z} = {T(w) , z} = [T(z) '] 2 {w, T(z) } .

8.6.10 Exercise. When {5, T} C Mo: a) If 5 has only one fixed point while T has two, then 5T -j. T 5; b) If 5 and T have the same fixed point ( s) , then 5T = T 5. c) If 5T = T 5 and each of 5 and T has only one fixed point , they share it .

8.6. 11 Exercise. a) If

and P � P (Z l�l , �2 ) �f z12 + L {

[Z _IW] 2

- �2 } ' then P, the Q 3w,tO

Weierstrafl elliptic function, is GW 1 ,W2 -automorphic. b) If f is GW 1 ,W2 -

automorphic and 5 � { a + tWl + SW2 : 0 :::; s, t < I } is the period parallelogram vertexed at a: bl) the sum of the residues of the poles of f in 5 is 0; b2) 2 :::; # [P(f) n 5] = #[Z (f) n 5] < 00.


8.6.12 Exercise. If

g2 �_ef 60 " 1 4 d clef " 1

� -------:- an g3 = 140 � 6 ' Q 3w,tO [Z - W] Q 3w,tO [Z - w]

then p' = 4p3 - g2P - g3 .

8.6.13 Exercise. If W �f mlWl + m2w2 -j. 0, then

367

8.6.14 Exercise. If el �f P (�l ) , e2 �f P (WI ; W2 ) , e3 �f P (�2 ) , then the ei , 1 :::; i :::; 3, are the three zeros of the polynomial function

Z r-+ 4z3 - g2Z - g3 .

8.6.15 Exercise. If a E Coo and multiplicities are taken into account, for P as in 8.4.19, # [p n p-l (a)] = 2 .

The statements in 8.6. 16 below are steps leading to the sufficient condition for the existence of a barrier at a point a in the boundary 8(Q) of a region Q, v. 6.3.29. The argument has been deferred to this part of the text because 8 .1 .8 is used.

8.6. 16 Exercise. If a E r �f 800 (Q) and no component of Coo \ Q consists of a alone: a) The assumption a = 00 is permissible. (Otherwise, for the map

1 ¢ : Coo '3 Z r-+ -- , z - a the argument may be conducted on ¢ (Coo ) . )

b) If C is a component of Coo \ Q and 00 E C, then Q1 �f Coo \ C is simply connected, d. 8.1 .8d) , and Q C Q l .

c) For some f in H (QI ) , exp[J(z)] = z, d. 8.1 .8h) (f is a determination of In) .

d) If Q2 �f f (Q) and L �f { a + it : a E lR, -00 < t < oo } , the line L meets Q2 in at most count ably many open line segments (w� , w�) of total length not exceeding 271" so that 'S (w�) > 'S (wU . (Otherwise, for some finite K, in Q2 there are K points Zl , . . . , ZK such that

K-l L [In (Zk+I ) - In (zk )] = 271"i, k=l

exp (In ZK - In zI ) = exp(271"i) = 1,


a contradiction.) e) If w �f u + iv, U 2: a there is a holomorphic function (h such that

w� - w w� - w

= exp [il'h(w)]

and 0 � fh (w ) � 7r. f) If

clef { -� '" 1'h(W) e(w) = 7r 7'

- 1

if �(w) 2: a

if �(w) < a

and Q3 �f { w : �(w) > a } , then

Furthermore, 2 7r -- arctan

( ) � e(w) � o. 7r � w - a

(The function f is a (holomorphic) determination of In. Hence the imaginary part of f is harmonic and so is (J. The calculation in e) and plane geometry provide the estimate . )

g) If an t 00 and en corresponds to an as e corresponds to a, then

00 (3(z) � L 2-nen (ln z)

n=O

defines a function subharmonic in Q and tending to zero as z -+ 00 . h) Near any point in r, for all sufficiently large n, the functions en take

on the value -l .

a . i ) In the sense of Ahlfors and Sario [AhS] , the function (3 i s a barrier at

[ 8.6.17 Note. The reason for g) is that e(ln z) can converge to o as z converges to some point on r .]

8.6. 18 Exercise. If Q is simply connected, then Q is a Dirichlet region. [Hint: If Q ¥ C, then Coo \ Q consists of precisely one component : 8.6.16 applies. If Q = C, 8(Q) = 0.]

9 Defective Functions

9.1. Introduction

The set E of entire functions is divided into two subsets, namely the set P of polynomial functions and T �f E \ P the set of transcendental functions.

If p E P, then p(C) = C is an abbreviated statement of the Fundamental Theorem of Algebra (FTA) . On the other hand, z -+ exp(z) is a transcendental function and exp(C) = C \ {O} .

If f E T, the Weierstrafi-Casorati Theorem (5.4.3c)) implies that near 00, the values assumed by f are dense in C. Thus zero is an isolated essential singularity of g : C \ {O} '3 z r-+ f (�) . By abuse of language, 00

is an isolated essential singularity of f. The following result embraces all the phenomena just described.

9.1 .1 THEOREM. (The Great Picard Theorem) IF a IS AN ISOLATED ESSENTIAL SINGULARITY OF A FUNCTION f, R > 0, AND f E H [A (a; 0, Rt] ' THEN # {C \ J [A(a; 0, Rt]} � 1 .

In every sufficiently small punctured neighborhood of an isolated essential singularity of a function f, the range of f omits at most one point.

The discussion is facilitated by the introduction of some special vocabulary and notation.

9.1 .2 DEFINITION. WHEN f E CIC , C \ f(C) �f D(f) IS THE SET OF defections OF f AND f IS # [D(f) ] -defective.

If f E E and f is 2-defective, say {a, b} C [C \ f(C) ] , then

] �f �= � E E and {O, I} C C \ [](C)] . Similarly, if f E M and f is 3-defective, say

{a, b, c} C [Coo \ f (Coo )] ,

369

370 Chapter 9. Defective Functions

- cl f C - b I - a then I � -- . --b E M and {a, 1 , 00} c [Coo \ I (Coo )] . The study of c - a 1 -2-defective entire resp. 3-defective meromorphic functions may be confined to functions with the simple sets of {0, 1 } resp. {0, 1 , 00} of defections. Hence, absent any further comment, for an entire 2-defective function I, D(f) = {o, I } ; for a meromorphic 3-defective function I, D(f) = {a, 1 , 00} .

The discussion below is devoted to showing first that 2-defective entire functions and 3-defective meromorphic functions are constants. Elaborations of those results provide the contents of the Great Picard Theorem.

clef 1 9.1 .3 Exercise. If I E M(C) and D(f) = {a, b, c} , then h =

I _ a E E

and D(h) = {_1_ , _1_} . (Thus the study of 3-defective meromorphic b - a b - c functions is reduced to the study of 2-defective entire functions. )

The arguments that follow are an amalgam of the efforts of several writers: Ahlfors, Bloch, Bonk, Caratheodory, Estermann, Landau, Minda, Montel, and Schottky. Picard's original proof of his Little Theorem is of an entirely different character, v. [Hil, Rud] .

Four steps are involved. The first seems irrelevant to the goal. A. (Bloch) If I E H (U) and 1 1' (0) 1 2: 1 , for some I-free positive constant

B, and some b (dependent on I) , B 2: 112 and I (1U) :J D(b, Bt.

B. I f h i s a nonconstant entire function and r > 0, for some b,

h (C) :J D(b, rt ·

The range of a nonconstant entire function contains open discs of arbitrary radius.

C. If l E E, I is not a constant , and D(f) = {a, I } , associated to I is a nonconstant entire function F such that

clef { � n7ri } D(F) :J S = ± In( Vm + V m - 1) + 2 : m E N, n E ;Z .

Since, for any b, D(b, It n S -j. 0, F(C) fails to contain any open disc of radius 1 , a contradiction of B: the Little Picard Theorem is valid.

D. Refined extensions, due to Schottky, of A together with a special application of the ideas behind the Arzela-Ascoli Theorem, v. 1 .6.9, lead to the Great Picard Theorem.

Section 9.2. Bloch's Theorem 371

9.2. Bloch's Theorem

The next rather general result provides an entry to the entire complex of Bloch/Landau/Schottky theorems.

9.2.1 Exercise. If X is a topological space, Y is a metric space, f : X r-+ Y is open, a E V E O(X) , and r5 �f inf { d[y, f(a)] : y E 8[J(V)] } , then r5 > 0 and B[J(a) , W C f(V) .

[Hint: 1 .3.7 applies.]

9.2.2 LEMMA. IF r > 0, f E H [D(a, r)] , M �f 1 1 !' I D (a,r) o 1 1 00 :::; 2 1!, (a) l , AND R = (3 - 2v2)r 1 !, (a) l , THEN D[f(a), R] O C f [D(a, rt] . PROOF. Consideration of the translate fr-a] (z) �f f(z - a) and the function fr-a] - f(a) shows that the assumption a = f(a) = 0 is admissible. If g(z) �f f(z) - !,(O) z and I z l < r, then

g (z) = r [!, (w) - !,(O)] dw, Jro,z] I g (z) l :::; I z l · 1

1 I !, (zt) - !,(O) I dt.

Cauchy's formula implies that when I w l < r ,

!,(w) - !,(O) = � r [ !'(u) _ !'(u) ] du = � r f'(u) du , 27rl J1u l =r u - w U 27rl J1u l =r u ( u - w)

I , , I Iw l M Iw l f (w) - f (0 :::; - ( I I ) ' 27rr = -1-1 M, 271' r r - w r - w

t I z l t I z l 2 1 I g(z) I :::; l z I Jo Mr _ l z l t dt = r _ l z l · 2 · M.

The triangle inequality implies I f(z) - f'(O)z l 2': I !, (O) I ' I z l - I f(z) l . Because � :::; I !, (O) I ,

I f(z) l 2': I !, (O) I ' I z l - Ig (z) 1 2': I !, (O) I ( I z l - r 1� I�Z I ) . (9.2.3)

Since I z l < r, the maximum value of the first factor in the rightmost member

of (9.2.3) is achieved when I z l = l1 � (1 - �) r « r) . The maximum

value itself is (3 - 2v2)r and 9.2 .1 applies when

X = Y = C, V = D(O, l1t, and r5 = ( 3 - 2v2)r I !, (O) I . D


9.2.4 Exercise. If f E H (U ) , f(O) = 0, 1' (0) = 1 , and R �f I l f l� l I oo , then f(1U) :J D(O, Rt · 1 [Hint: 3 - 2v2 > - .] 6 9.2.5 Exercise. a) If f E H (U ) , then '1jJ : U '3 z r-+ I I' (z) 1 (1 - I z l ) IS continuous. b) If

IT 1C ( ) clef clef ( 3 In) a E \U , '1jJ (a) = max'1jJ z = M, and R = - - V 2 M, zEllJC 2

then f(1U) :J D(f(a) , Rt . c) R > I f��) 1 � 112,

l - I a l [Hint: M = (1 - l a l ) I I' (a) 1 and if I z - a l < -2-' then

I I'(z) 1 :::; 2 1 1'(a) 1 ; 9.2 .1 applies.]

It is a short step from 9.2.5 to A: b = f(a) and f(1U) :J D (b, 112 )

[ 9.2.6 Note. A heuristic (but invalid) argument for A is the following.

If 1 1' (0) 1 � 1, for z near 0, I f(z) - f(O) 1 � 1; 1 . Thus, if r is small and positive, each point in the Jordan curveimage Jr �f f[Go (r)] (the image under f of the perimeter of the open disc D(O, rt) is at a distance not less than r 2" from 0 [= f(O)] . Thus

f(1U) :J J [D(O, r)O ] = U J [Go(s) ] O�s<r

:J U Gf(O) G) = D [f(O) , �] ° . O�s<r

Even if the display above is valid, it does not imply A, since there r is no lower bound on "2 ' i.e. , there is no implication that

f(1U) :J D [1(0) , 112r

enz - 1 Indeed, if f(z) = , then f(O) = 0, 1' (0) = 1 ; if n is large, 1 n [ 1 ] 0

- - tic f(q , which shows that if n is large, f(1U) 1; D f(O) , -n 12

Section 9.2. Bloch's Theorem

Thus, although 1 (1U) contains some D (b, 112 ) 0 , b is not neces

sarily 1(0) (rather, for a as found in 9.2.5, b = I(a) . ]

9.2.7 Exercise. The assertion in B is valid. [Hint: If M > 0, for some a, I h'(a) 1 > M. Translations and res calings apply.]

9.2.8 THEOREM. IF R > 0, 1 E H [D(O, R)] , AND

373

{0, 1 } c C \ I [D(O, Rr] , (9.2 .9)

THEN: a) FOR SOME FUNCTION F IN H [D(O, Rr] '

I(z) = - exp[7ri cosh 2F(z) ] ;

b) A VALUE OF F(O) CAN BE DETERMINED BY REFERENCE TO 1(0) ; c) THE RANGE OF F CONTAINS NO OPEN DISC OF RADIUS ONE, i.e. ,

{b E q '* {D(b, 1r � F [D(O, Rt] } .

PROOF. a) The truth of each of the following statements is implied by 8.1 .8 and the hypotheses:

• for some h in H [D(O, Rr] ' 1 = exp (27rih) ; • for some u in H [D(O, Rr] ' h = u2 ; • for some v in H [D(O, Rt] ' h - 1 = v2 ;

2 2 1 • u - v = 1 and u - v = -- ¥= 0;

u + v • for some F in H [D(O, Rt] ' u - v = exp(F) ; • I ID(O,R) O = - exp( 7ri cosh 2F) .

b) If

{k, m} C IZ,

2 ln 1 1(0) 1 + \2k - l)7ri �f (3 7rl

In 1 (3 ± �I + 2m7ri F(O) �f

2 '

then 1(0) = - exp [7ri cosh 2F(0) ] . c ) If

clef { � n7ri (T E L = ± In (Vm + v m - 1) + T m E N, n E IZ }


and, for some z in D(O, Rt , F(z) = a, then I(z) = 1 , a contradiction of (9.2.9 ) . The remainder of the argument consists of showing that L meets every open disc of radius one.

The estimates

In(vm + 1 + vim) - In(vIm + vm - 1) { = In(h + 1) < 1 m + 1 < In J :::; In y'3 < 1 m - 1

imply that if a E C, for some a in L,

if m = 1 if m > l '

1 y'3 I�(a) - �(O') I < 2 ' I CS(a) - CS(o') 1 < 2 '

whence l a - 0'1 < l .

9.2.10 Exercise. The PROOF above is valid if 1 is entire.

9.3. The Little Picard Theorem

The preceding developments lead to the final statement in C.

D

9.3.1 THEOREM. (The Little Picard Theorem) a) IF 1 E E AND 1 IS 2-DEFECTIVE, 1 IS A CONSTANT. b) IF g E M(C) AND g IS 3-DEFECTIVE, g IS A CONSTANT.

PROOF. Since 9.1 .3 reduces b) to a) , an argument for a) suffices. Associated with 1 is F of 9.2.8. If 1 is not constant , neither is F. By virtue of B, (v . 9.2.7) , on the one hand, F (C) contains arbitrarily large open discs, yet contains no open disc of radius one, a contradiction. D

[ 9.3 .2 Note. The derivation above of the Little Picard Theorem uses only the existence of some positive constant B for which A obtains. Some interest attaches to the determination of the supremum B of all such B.

For .1' � { I : 1 E H (1UC ) , ! ' (0) = I } , and 1 in .1',

Af � sup { R : 1(1U) contains some D(b, Rt }

and A �f sup A f . If 1 in .1', then 1 is injective on some subregions fEF Q ofU For a subregion Q on which 1 is injective, Rf,n is the radius of the largest open disc contained in I(Q) . Then Bf �f sup Rf,n nel[]

Section 9.3. The Little Picard Theorem

B clef

B � clef and = sup f . Finally, if 1 E F = F n { I 1 is injective } ,

f then

� { � Af = sup R 1(1U) contains some D(b, Rt } and A = sur: Af .

fEF

For the various constants above, the best estimates known to the writer are:

7r 0.433 + 10- 14 < B < 0.472; 0.5 < A < 0.544; 0.5 :::; A :::; 4 '

The variational aspect of the PROOF of Riemann's Mapping Theorem (v. 8.1 . 13) suggests that a similar technique is useful in the derivation of A. In fact a variational approach yields

A 2: 3� - 2 (� 0.121320343) ,

3 which is far better than A 2: "2 - v2 (� 0.085786437) as found above. Ahlfors, via an improvement of Schwarz's lemma, showed

B 2: V; (� 0.433012701 ) . He and Grunsky conjectured, but did

not pmve B � J va - 1 r m r (H) , 2 r (�) (� 0.4719) .]

375

9.3.3 Exercise. 1 . a) B :::; A :::; A; b) k : 1U '3 z r-+ "2 [In(l + z) - In(l - Z)] 1S

- 7r in F; c) A :::; 4 ' clef 9.3.4 THEOREM. IF 1 E E AND g = 1 0 1, EITHER g HAS A FIXED-POINT

OR I(z) = z + b AND b -j. O.

PROOF. If g has no fixed-point neither has I. Thus

g(z) - z k : C '3 z r-+ -'--':---C.-_ I(z) - z

is entire. If k (c) = 0, then g(c) = c, a contradiction. If k(c) = 1 , then g(c) = I(c) , i .e . , 1[J(c)] = I(c) , whence I(c) is a fixed point of I, a second


contradiction: {O, I} C C \ k(C) . The Little Picard Theorem implies that for some c in C, k == c, whence !, (I' 0 I - c) == 1 - c. Thus

Z (I' 0 I - c) u Z (I') = (/)

and {O, c} C C \ !' 0 I(C) . The Little Picard Theorem implies that for some d in C, !' 0 I == d. Hence !, is constant . D

9.3.5 Exercise. The function h : C E z r-+ z + eZ is entire and has no fixed-point . What is the set of fixed-points of h 0 h?

9.4. The Great Picard Theorem

Bloch's Theorem makes no direct mention of defective functions. On the other hand, the burden of its message is that a function I in H (U ) and normalized by the condition I !' (0) I 2: 1 is locally not defective. The next and related result incorporates 2-defectivity in the hypothesis.

9.4. 1 THEOREM . (Schottky) . IF 0 � fJ < 1, I E H (U ) , 1(0) = a, AND I IS 2-DEFECTIVE, FOR SOME FUNCTION

<t> : [2, (0) x [0, 1 ) '3 (w, fJ) r-+ <t> (w, fJ) E (0, 00) ,

{ { I a l � w} 1\ { I z l � fJ}} '* { 1 1(z) 1 < <t>(w, fJ) } .

[ 9.4.2 Remark. Thus the absolute value of a 2-defective function I in H (U) is, for some given bound on the value of 11(0) 1 (= l a l ) , bounded in D(O, fJ ) by an I-free constant .]

PROOF. The notations used below are those in the PROOF of 9.2.8. The 1 1 function h may be chosen so that - 2 < �[h(O)] � 2 '

1 1 I ln l a l l 1 In w clef If l a l 2: :; , then I h(O) 1 � 2 + � � 2 + 271" = PI (w) and the fol-lowing estimates emerge:

lu(O) 1 = VTh(6)T < 2VTh(6)T �f P2(w) ,

Iv(O) 1 = Vl h (O) - 1 1 < 2vl h(0) - 1 1 � P3 (w) , lu (O) - v (O) 1 < 2 Iu(0) - v(O) 1 �f P4(w) ,

1 clef I ( ) I

= lu (O) + v(O) 1 < 2 Iu(0) + v (O) 1 = P5 (W) . u 0 - v(O)

The function F (in H (1U) ) may be determined so that -71" < 'S[F(O)] � 71". Then

IF(O) I � l In l u(O) - v(O) 1 1 + 71" < 2( l ln lu(O) - v(O) 1 1 + 71") �f P6(W) .

Section 9.4. The Great Picard Theorem 377

If I � I :::; fJ and F'(�) -j. 0, then ( : 1U '3 z r-+ Fg � ��;,(i)z] is in H (1U) and 1 ( (1U) contains no open disc of radius ( 1 _ fJ) IF' (�) I ' Since ('(0) = 1, 9.2.4

implies

IF' (0 I < 1 : fJ ' (9.4.3)

As derived, (9.4.3) is valid because F'(�) -j. 0; the conclusion holds a fortiori if F' (�) = O. Integration implies

6 6 IF(z) - F(O) I :::; I _ fJ · fJ l (a, fJ)

I I(z) 1 < exp [7r cosh 2<t>1 (a, fJ)] �f <t>(a, fJ) .

1 1 1 If l a l < =.; ' then l a l :::; "2 and 2 :::; 1 1 - a l :::; 2. The conclusion in the preceding paragraph applies to 1 - I. D

9.4.4 Exercise. The hypothesis w 2': 2 may be replaced by w 2': O. 9.4.5 Exercise. a) Schottky's Theorem implies Bloch's Theorem. b) Schottky's Theorem implies Landau's Theorem, viz . :

For some map ¢ : C '3 z r-+ ¢(z) E (0, 00) , {J E H [D(O, ¢( a)] } ::::} {J i s not 2-defective} .

c) Landau's Theorem implies Bloch's Theorem.

9.4.6 DEFINITION. A SUBSET F OF Cn (VIEWED AS A SUBSET OF C� ) IS Coo -normal IFF EVERY SEQUENCE {In}nE]\/ CONTAINED IN F CONTAINS A SUBSEQUENCE THAT IS LOCALLY UNIFORMLY CONVERGENT IN C� .

9.4.7 THEOREM. (Montel) THE SET F �f { I : 1 E H (fl) , D(f) = {O, I} } IS Coo-NORMAL.

PROOF. If a E fl and

Fa,M �f { I : 1 E F, I I(a) 1 :::; M < oo } , (9.4.8)

translations and changes of scale permit the assumptions :

a = 0, 1Uc c fl, and M = 1 . (9.4.9)


Schottky's Theorem implies that if 0 :::; fJ < 1 , then

sup { 1 I I I D (O,l!) 1 1 00 : I E Fa,M } :::; <1>(I , fJ) < 00 .

Thus, for some neighborhood W of zero, e.g., W = D(O, fJt, if w E W and I E Fa,M , then I I(w) 1 :::; <1> (1 , fJ) . When restrictions (9.4.9) are abandoned,

the conclusion is: V �f { z : z E Q, sup I I(z) 1 < oo } is an open subset JEF of Q. Furthermore, a E V, whence V -j. 0.

If Q -j. V and b E Q \ V, then { 1 1(b) 1 : I E F } is not a bounded set of numbers. Thus there is a sequence {In} nE]\/ contained in F and such that

I ln(b) 1 t 00. The sequence { II �f gn} is contained in F and Schot-n nE]\/ tky's Theorem implies that for some open neighborhood U of b and some positive P, {z E U} ::::} { lgn(z) 1 :::; P} . Vitali's Theorem (5.3.60) implies that for a subsequence, again denoted {gn}nE]\/' and a g in H (Q) , the gn converge uniformly on each compact subset of U to g. Since lim gn (b) = 0,

n--+ CXJ 5.4.39 implies a) g == 0 or b) 0 tt g(Q). If a) is true, for every z in Q,

lim In(z) = 00 n---+oo whereas, if z E V, lim 1 1,, (z) 1 < 00 , a contradiction. Thus a) is false n---+oo and b) is true. Consequently lim gn(b) -j. 0, a second contradiction. In-n---+oo eluctably, Q \ V = 0.

If Q contains an a such that (9.4.8) obtains, then Q = V, i .e.,

{z E Q} ::::} { sup I I(z) 1 < oo} . JEF Then 5.9.13 implies F is normal. In particular, if

then { l ln (a) 1 : n E N } is a bounded set of (real) numbers . There remains the possibility that for each a in Q, (9.4.8) is false.

Hence, for each a in Q, and any sequence {In} nE]\/ contained in F,

{ l ln (a) 1 : n E N } . b d d Th { 1 clef } . . d ' F d . IS un oun e . e sequence - = gn IS contalIle In an , VIa In nE]\/ passage to a subsequence as needed, the earlier argument shows this time

Section 9.4. The Great Picard Theorem 379

that g == O. Thus {In}nE]\/ converges to 00 on every compact subset of Q: :F is Coo -normal. D

9.4.10 THEOREM. (The Great Picard Theorem) IF I E H [A(a; O, rt] , a IS AN ESSENTIAL SINGULARITY OF I, AND 0 < s < r, THEN

# {C \ I [A(a; O, st ] } �f N :::; 1 .

PROOF. If N > 1 , i t is permissible to assume

a = 0, D(f) :J {O, I } , r = 1 .

Then 9.4.7 implies that the sequence

{In : A (0; 0, It '3 Z r--+ I (�) } nE]\/ is Coo-normal. Hence, via passage to a subsequence as needed, either for some positive M and every n , In [C� (O)] c D(O, M) or, in the notation

used in 9.4.7, gn [C � (0)] c D(O, M). Passage to the limit yields

or g [C2� (0)] c D(O, M) . The Maximum Modulus Theorem implies that

for each n in N, I [A (0; n : 1 ' �] 0) c D(O, M) or

1 Hence I I(z) 1 or I I(z) 1 is bounded near zero, which is impossible because zero is an isolated essential singularity of I· D

9.4. 11 COROLLARY. IF c IS AN ISOLATED ESSENTIAL SINGULARITY OF I, THERE IS AT MOST ONE COMPLEX NUMBER a SUCH THAT IN EACH NEIGHBORHOOD N(c) , I(z) = a FAILS TO HAVE INFINITELY MANY SOLUTIONS. PROOF. The argument given above applies to each N (c) . Hence, if I(z) = a has no solution in N (c) and b -j. a, then I(z) = b has a solution Z1 in N(c) . If

1 1 Z1 tt 2 N ( c) , then I (z) = b has a solution Z2 in 22 N (c) . Induction provides a sequence {zn}nE]\/ converging to c and such that I (zn) = b. D


9 . 5 . Miscellaneous Exercises

9.5.1 Exercise. If 1 and g are entire and ef + e9 = 1, then 1 and g are constant functions .

[Hint: In the equation ef = 1 - e9 the left member is never zero. Hence {2mri} nET- C [C \ g(C)] .]

9.5.2 Exercise. If 1 and g are entire and 12 + g2 = 1, for some entire function h, 1 = cos oh and g = sin oh.

[Hint: In the equation 12 + g2 = (f + ig) . (f - ig) neither factor of the right member can be zero. Then 8 .1 .8h) applies.]

9.5.3 Exercise. If 1 E T, there is at most one complex number a for which I(z) = a does not have infinitely many solutions .

[Hint: If g(z) �f 1 (�) , z -j. 0, zero is an isolated essential sin

gularity of g.] 9.5.4 Exercise. If zero is an isolated essential singularity of I, for some c in A(O; 0, 1 ) and for some a in C, if ° < E < l e i , then

9.5.5 Exercise. If 1 is entire and periodic, there are infinitely many 1-fixed-points.

[Hint: If g(z) �f I(z) - z, then g is entire. If p is a nonzero period of I, for all but at most one n in Z, np E g(C) .]

9.5.6 Exercise. There are non defective non polynomial entire functions . 9.5.7 Exercise. If x E [0, 00] , there is a nondefective entire function such that p(f) = x (v. 7.3.3) .

1 0 Riemann Surfaces

10.1. Analytic Continuation

A fundamental fact about a function 1 in H (fl) is that for each a in fl, there is a (unique! ) sequence {cn,a} nEZ+ and in (0, 00] some radius of convergence Ta such that if I z - a l < Ta , then

00 I(z) = L cn,a (z - a)n

n=O (v. 5.3.23) . If I b - a l < Ta , the equation

= 00 L cn,a [z - b + (b - a)] = L Cn,b (Z - b)n n=O n=O

serves to define uniquely the sequence {Cn,b} nEZ+ ' 10.1.3 Exercise. In (10. 1 .2 ) ,

( 10. 1 . 1 )

(10. 1 .2)

(10 .1 .4)

The series in the right member of (10 .1 .4) converges. For the radius of convergence Tb of the right member of (10. 1 .2 ) , there obtains the inequality: Tb 2': Ta - I b - a l · 10.1 .5 Exercise. The function T : C '3 b r-+ Tb E Coo is continuous.

When Ta = 00, the right member of (10. 1 . 1 ) defines a function F holomorphic in C, i .e., F E E, and F ln= I. The Identity Theorem (5.3.52) implies that if G E E and Gln= I, then F = G: F as described is unique.

On the other hand, if Ta < 00, the discussion takes a more extended form. The following Examples offer a sampling of the variety of phenomena to be encountered. 10.1 .6 Example. If fl �f C \ (-00, 0] and Z E fl, then [1, z] e fl and the

. clef 1 dw equatlOn I(z) = - defines a function in H(fl) . Hence, if a E fl, then [l ,z] w

381

382 Chapter 10. Riemann Surfaces

� J(nl (a) clef � J(Z) = � I (Z - a)n = � Cn(Z - a)n and Ta = l a l . Furthermore, n. n=O n=O if z E Q, then I'(z) = � . If F(z) �f exp[J(z)] , then [F�Z) r = O. Since

J(I) = 0, F(z) = z, i.e. , exp[J(z)] = z, J(z) is a branch of In z. If a = 1 and z E D(I , It , then

J(z) = r dw = r dw J[l , Z] w J[l , Z] 1 + (w - l)

= fl (-I )n (w - l)n dw n=O [l ,z]

(Xl ( It+1 (Xl = 2:(-1)" z - �f 2: Cn (z - lt �f h(z) . ( 10.1 .7) n=O n + 1 n=O

Since Tl = 1, (10. 1 .7) provides no definition of h outside D(I , It . Nevertheless, for the Junction elements , i.e. , the pairs (I, Q) and [h , D(I , It ] , the following obtain .

• Q n D(I , lt -j. 0; • J lnnD( l , l l o = h lnnD(l , l ) O : J and h agree where they are both defined and J provides a natural extension of h to points outside D(I , lt · These circumstances motivate

10.1.8 DEFINITION. FUNCTION ELEMENTS (h , Qd AND (12, Q2) ARE immediate analytic continuations OF ONE ANOTHER IFF: a) Ii E H (Qi ) , i = 1 , 2 ; b) Q1 n Q2 -j. 0; c) h I n1nn2 = 12 I n1nn2

WHEN f E N, A FUNCTION ELEMENT (II , QJ ) IS AN analytic continuation OF A FUNCTION ELEMENT (h , Qd IFF THERE IS A FINITE SEQUENCE OR chain {(Ii, Qi ) } l �i�J SUCH THAT (Ii, Qi) IS AN IMMEDIATE ANALYTIC CONTINUATION OF (Ii- I , Qi-d , 2 � i � f, .

When ambiguity is unlikely, the function II itself is called the analytic continuation of h .

The roles played by (h , Qd and (12, Q2 ) are symmetrical. Each of h and 12 can be an analytic continuation of the other at points where the other is not yet defined. In 10.1.9-10.1 .12 below, J is the function in 10.1.6 and

Q = C \ (-00 , 0] .

Section 10. 1. Analytic Continuation 383

10.1.9 Exercise. If Z E fl, the closed interval [1 , z] may be covered by a finite set of open discs Di �f D (Zi , rit , 1 � i � I, such that each is contained in fl and Z1 = 1 , zi+1 E Di , 1 � i � I - 1 . In that event , the power series representation for I I D is the result of the next equations, i + l

(Xl I(n) (Zi ) n (Xl 1(") (Zi ) n L , (z - Zi ) = L , [z - Zi+1 + (Zi+1 - Zi )] n . n. n=O n=O

v. (10 .1 .2 ) .

(Xl I(n) ( ) _ ""' zi+1 ( _ . )n - � , Z Z,+1 , n. n=O

10. 1 .10 Exercise. If Z �f x + iy E fl and x is fixed, then

Furthermore,

:3 lim I (z) �f L + (x) and :3 lim I (z) �f L - (x) . y+O yto

if x < 0 if x = 0 . if x > 0

10. 1 . 1 1 Exercise. There is no entire function F such that F ln= I. 10. 1 .12 Exercise. a) If g �f exp (�) , then g E H (fl) and g(z) 2 = Z in

fl. b) There is no entire function G such that Gln= g. c) If

h(z) �f Vi = )1 + (z - 1) = 1 + f (� - 1) (� -�? . . . (� - Tn) (z _ 1) Tn �f f Cm (z _ 1) Tn

'

Tn=1 n=O

then r1 = 1. If D(a, rr c fl, then [g, D(a, rr] is an analytic continuation of [h, D(I , lr] .

Despite the conclusions in 10. 1 . 10 and 10.1 . 11 , i f a �f a + ib E fl and

(Xl (n) ( (Xl I( ) �f ""' I a ) ( _ )" �f ""' ( _ )" Z - � , Z a - � cn Z a , n . n=O n=O (10. 1 . 13)

then ra = 1 0' 1 ; if c < 0, then D (a , rat n (-00, 0] i- 0: D (a , rat is not contained in fl. Consequently, k denoting the function represented by the series in (10. 1 . 13 ) , if z E D (a , rat and 'S(z) < 0, then k(z) i- I(z) , even though [k, D (a , rat] is an analytic continuation of [h , D(I , lr] .


In D (a , rat n { z of [h, D(l , 1rl .

'S( z) < O } there are two analytic continuations

The process of re-arrangement of series can (sometimes) yield an analytic continuation [k, D (a , Rc)O l of [h, D(l , 1rl that is different from (I, Q) .

10.1 .14 Example. If f E H (Q) , a E Q, and J' (a) -j. 0, the Inverse Function Theorem, 5.3.41 , implies that for some positive r, in H {J [D (a , rrn there is a g such that g 0 f I D(a,r) O == z. Furthermore, [g , D( a, rrl engenders analytic continuations to other function elements. Local inverse functions like g above, the local inverse of exp 0 exp, and sin constitute a particularly rich source of analytic continuations exhibiting the phenomena described above. In fact , 10.1.6 is based on the local inverse of the entire function z r-+ exp(z) .

10.1 .15 DEFINITION. WHEN f E H (Q) , a E 8(Q) , r > 0, a IS A regular point OF f IFF SOME [g , D( a, r rl IS AN IMMEDIATE ANALYTIC CONTINUATION OF (I, Q) . WHEN b E 8(Q) AND b IS NOT A REGULAR POINT OF f, b IS A singular point OF f.

The set of regular points of f may be empty, e.g. , if Q = C (in which case the set of singular points is also empty) or if f is such that 8(Q) is the natural boundary for f (in which case the set of singular points is 8(Q) . )

A singular point a of f i s isolated iff f i s represented by a Laurent 00 series, L cnzn and for some negative n , Cn -j. O.

n==-(X) However, if S �f {O} U { z : mrz = I } and

) clef . ( 1 ) f(Z = sm -; ,

g( z) �f { cosc (� ) if z tt S

otherwise

then {J, g} c H (C \ S) , each point of S \ {O} is a pole of f and an isolated essential singularity of g and 0 is a nonisolated singularity of both f and g.

00 10.1 .16 THEOREM. (Pringsheim) IF EACH cn IN L cnzn IS NONNEGATIVE

n=O 00 AND Rc = 1 , THEN 1 IS A SINGULAR POINT OF f : 1U '3 z r-+ L cnzn .

n=O

Section 10. 1. Analytic Continuation

PROOF. Since zn = � ( n ) (z - �) Tn , the re-arrangement � m 2n-Tn Tn=O 385

(10. 1 . 17)

00 of L cnzn diverges if I z l > 1 . Owing to the hypothesis Cn 2': 0, n E N+ ,

n=O Fubini's Theorem (4.4.9) implies that if 0 < z < 1, the inversion of the order of summation is valid in (10 .1 . 17) :

� '"z" � � [t>(;,) (Z2:_�

�

l

= �O Cn [� (:) 2n�Tn 1 (z _ �) Tn

� � !(�:,�D (z - D�

(10. 1 . 18)

If 1 is not a singularity of f, for some positive r there is a function element [g , D(I , rtl that is an immediate analytic continuation of (I, 1U) . Hence

1 the distance d between "2 and some point of the boundary of 1U U D(I , rt 1 exceeds r the right member of ( 10 . 1 . 18) converges for some z in ( 1 , 00 ) , a

contradiction. D

10.1.19 Exercise. Pringsheim's Theorem (10. 1 . 16) obtains if the hypothesis Cn 2': 0, n E N+ , is replaced by � (cn ) 2': 0, n E N+ , and it is as-

00 00 sumed that the radii of convergence of both L enzn and L � (en ) zn are one. n=O n=O

00 00 g (Tn) (�) (

I ) Tn [Hint: For g : 1U 3 z r-+ L � (cn ) zr> ' L , z - -m. 2 n=O Tn=O diverges if z > 1. Furthermore, g (Tn) (�) = � [f(Tn) (�) ] and

f(Tn) ( 1 ) � m! "2 ( z _ �) Tn


diverges if z > 1 (whether

converges or diverges when z > 1) . ]

10. 1 .20 THEOREM. IF THE RADIUS OF CONVERGENCE Re OF =

P(a, z) �f L cn (z - a)n n=O

IS POSITIVE (AND FINITE) , FOR SOME b IN ea (Re ) , NO FUNCTION ELEMENT [g, D(b, sr] IS AN IMMEDIATE ANALYTIC CONTINUATION OF

[P(a, z) , D (a , RetJ , i.e. , SOME POINT b OF ea (Re ) IS A SINGULAR POINT OF

00 I : D (a, Ret r-+ L en (z - a)n .

n=O

PROOF. Otherwise, each b of ea (Re ) lies in an open annular sector

and there is a function element (jb, Ab ) that is an immediate analytic continuation of [I, D( a, rr] . Since ea (Re ) is compact , ea (Re ) contains a £1-

M nite set {bmL<_m<_M such that U Ab

= :J ea (Re ) . If s = max rb and 1 �m�M = m=1

5 = min Rb , then s < Re < 5 and 1�m�M =

D (a, 5) c D (a, Re ) U A(a; s, 5) .

If z E D (a, Ret n Abk n Ab" the Identity Theorem implies Ibk (z) = Ibl (z) . Hence the radius of convergence of P(a, z) is at least 5, a contradiction.

D

d f ex:> zn 10.1 .21 Example. The radius of convergence of P(O, z) � L n2 is one.

n=l Furthermore P(O, z) converges for each z in T. However, 10.1 . 16 implies that 1 is a singular point of I : 1U '3 z r-+ P(O, z) .

Section 10. 1. Analytic Continuation 387

10.1 .22 Example. The opposite extreme of a function element (I, Q) that permits analytic continuation beyond Q is a function element (I, Q) such that 8(Q) is a natural boundary for f (v. 7.1 .28) .

00 10.1 .23 Exercise. The radius of convergence of L zn! is 1. The se-

n=O ries defines a function f in H (1U) . Furthermore, if q E Qi, ° � r < 1, and z = re27rqi , then I f (z) I t 00 as r t 1. If F is a function holomorphic in a region Q meeting 1U and F lnn1U= f lnn1U , then Q C 1U: '][' is a natural boundary for f. 00

S· ·l 1 · b · C ( ) clef ,", 2" Iml ar conc USlOnS 0 tam lor g z = � z . n=O

The phenomena in 10.1.23 exemplify the next two results.

PI < P2 < . . . ;

c) 1 < M E N; d) qn > (1 + � ) Pn , n E N; e) FOR � Cnzn REPRESENT

ING f, Rc = 1; f) 1 IS A REGULAR POINT OF f; AND g) Cn = 0 WHEN Pk < n < qk , k E N, THE SEquENCE {Spk } kEN OF PARTIAL suMS OF

00 L Cnzn n=O

CONVERGES IN A NONEMPTY NEIGHBORHOOD OF 1 .

PROOF. Because 1 is a regular point of f , it follows that some analytic continuation of f is holomorphic in some Q containing 1U U {I} . For the entire function defined by g (z) �f � (zM + zM + 1 ) :

2 • g(l) = 1 ; • if z E D(O, 1) \ { I} , then 1 1 + z l < 2, and

1 I g(z) 1 = 2 1 zM I · 1 ( 1 + z) 1 < 1 .

Thus K �f g[D(O, 1) ] C 1U U {I} C Q: the compact set K i s a subset of the open set Q. Hence, for some positive 15 ,

O(C) '3 U �f U D(z, r5t C Q. zEK


The uniform continuity of g on D(O, l ) implies that for some positive E, g [D(O, 1 + Et] c Q. Since h(z) �f 1 0 g(z) is holomorphic in g- l (Q) , it follows that for some

00 {bn}nEl\! and a positive E, h(z) = L brnzrn obtains in D(O, 1 + Et . Direct

rn=O calculation shows that for k in N,

Pk (M+l )Pk SPk (z) = L cn[g(z)t = L brnzrn .

n=O rn=O (10. 1 . 25)

The rightmost member of (10. 1 .25) converges in D(O, 1 + E) O , i.e., {Spk } kEl\! converges in D(O, 1 + E)O as required. D

10.1 .26 THEOREM . (Hadamard) IN THE NOTATIONS AND CONDITIONS

OF 10.1 .24, IF ck i- 0, k E N, Pk+l > (1 + �) Pk , I(z) = � CkZPk , AND

Rc = 1, THEN '][' IS A NATURAL BOUNDARY FOR I. PROOF. If a E '][' and a is a regular point of I, then 1 is a regular point of I(O'z) . Hence the argument in 10.1 .24 applies: {sPk (z)}kEl\! converges in some D(O, 1 + Et . In the current circumstances, {spk hEl\! = {sp}PE]\/" Hence, if a is a regular point of I, in some D(O, 1 + Et , {Sp(z)} pEl\! con-

00 verges, i .e. , in some D(O, 1 + Et , L cnzn converges. This conclusion con-

n=O tradicts the assumption: Rc = 1 . D

[ 10.1.27 Note. In 10.1 .24, although the series representing 1 diverges for each z outside D(O, 1 ) , some sequence of partial sums of the series does converge in some nonempty neighborhood of 1 : there i s overconvergenCE. By contrast , for any sequence {Spk } kEl\! of partial sums of the

00 series L zn! , if q E Qi, R > 1 , and z = Re27rqi , then I Spk (Z) 1 t 00:

n=O overconvergence is absent . In 10.1.24 the sizes of the gaps [the sequences of successive zero coefficients] increase rapidly, while the sizes of the nongaps [the sequences of successive nonzero coefficients] may increase as well. Hadamard's result asserts that if the size of each nongap is one and the sizes of the gaps increase sufficiently rapidly, the boundary of the circle of convergence is a natural boundary. What follows is an illustration of what can happen when the gaps are Hadamard-like per the hypotheses of 10.1.26 and the sizes of the nongaps also increase sufficiently rapidly.

Section 10. 1 . Analytic Continuation

If P(z) �f Z (Z + 1 ) , then

00 (10. 1 .28)

converges and defines a function f holomorphic in each component of Q �f { z : IP(z) 1 < I } . A point a is on the boundary of a component of Q iff IP(a) J = 1 . One component , say C, contains, for some maximal positive r, D(O, rt. Owing to 10.1 .26, the boundaries of the components of Q are natural boundaries for the re-strictions of f to those components. Since S �f Co(r) n 8(C) "I- 0, if a E S, then a is a singular point of f. In D(O, rt, f is represented by a power series P(O, z) calculable from ( 10. 1 .28) . Furthermore a given power of z appears in at most one term of (10. 1 .28) . Hence both the gaps and the nongaps in P(O, z) are Hadamardlike. Overconvergence occurs for some z in Q \ D(O, r) ("I- 0) .

ex:> 2n If H (z) �f L �, the radius of convergence of the right member n n= l is 1 . Owing to 10.1 .26, '][' is a natural boundary for h. On the other hand, whereas I f I and I g l in 10.1 .23 are unbounded in 1U,

ex:> 1 7[2 if I z l < 1 , IH(z) 1 ::::: L n2 = 6· n=l If I z l < 1 , then f(z) �f _1_ = � zn and the radius of converz - 1 � n=O gence of the right member is 1, while 1 + iO is the unique singular point of f. Nevertheless, if I z l > 1 and {snk (z)}kE]\/ is a sequence

00 of partial sums of L zn, {Snk (z) }kE]\/ diverges : overconvergence

n=O is absent .

There is an extended discussion of the phenomena noted above in [Di] .]

389

For the functions f resp. g in 10.1 .6 resp. 10.1 .12 , if (II , Q) resp. (gl , Q) is an analytic continuation of [J, D(I , 1 t] resp. [g , D(I , It] ' the Identity Theorem implies that in Q, eh (z) = z resp. gl (z)2 = z. The three results that follow elaborate on this theme.

10.1 .29 Exercise. For the polynomial

Tn l = l , . . . ,Tnn= l a w Tn 1 . . . wTnn 'TTl l , · · · , 'TTln 1 n ,


if { (II , Q) , . . . , Un , Q)} is a set of function elements such that on Q,

P (II , · · · , fn ) = O,

for any set { (gl , QI ) , . . . , (gn , QI ) } of function elements that are analytic continuations of the { (II , Q) , . . . , Un , Q) } , the equation

is valid on QI . [Hint: The Identity Theorem for holomorphic functions applies. ]

10.1 .30 Exercise. If, in the context of 10.1 .29, f is a solution on Q of the differential equation P (y, y' , y", . . . ) = 0 and (g, QI ) is an analytic continuation of (I, Q) , on QI , P (g, g' , g" , . . . , ) = O. 10.1 .31 Exercise. The conclusion in 10.1 .29 remains valid if P is re-CXJ , • • • ,CXl placed by a power series L a'ml , • • • . 'mn W�l • • • w;:'n that converges

'TTl l= l , ·· · ,'TTln=l in a nonempty polydisc X:= I D (0, r k ) 0 •

[ 10.1.32 Remark. The phenomena in 10.1 .29-10.1.31 are examples of the Permanence of Functional Equations (under analytic continuation) .

Thus , although analytic continuation permits the creation of new function elements from old, when 1 � m � n, (I'm , Q) resp. (grn , Q) are analytic continuations of one another, and

(in this instance, the second members of all the fUllction elements are taken to be the same) , nevertheless, functional relations among the f'm ' 1 � m � n, persist among the grn , 1 � m � n .]

10.1 .33 Exercise. In 10.1.29 , if n = 2 and P (WI , W2 ) �f wi - w� , then (II , QI ) � (z r-+ z, q, (12, Q2 ) �f (z r-+ -z, q are function elements such that P (II , II ) = P (12, h) = O. Yet neither of (II , QI ) and (12, Q2 ) is an analytic continuation of the other.

The converse of the principle of the Permanence of Functional Equations (under analytic continuation) is false.

Section 10.2. Manifolds and Riemann Surfaces

10.2. Manifolds and Riemann Surfaces

The developments that follow organize the study of functions such as

In : D ( 1 , 1 t '3 z r-+ In z E C, vrD(I , It '3 z r-+ Vi E C,

391

and other functions afflicted with ambiguity (multivaluedness) when their original domains are extended, e.g. , to C \ {a}.

As the discussion in Section 10.1 reveals, a power series P( a, z) can engender analytic continuations to power series PI (b, z) and P2 (b, z) such that PI (b, z) -j. P2 (b, z) . The definition of the word function as it is used in mathematics makes the term multivalued function an oxymoron, although there is a temptation to describe as multivalued a function f that is locally represented by P( a, z ) , PI (b, z ) , and P2 (b, z ) .

Further discussion can be conducted systematically in the context established by the following items.

• A complete analytic function is a collection CAF �f {(Iv , flv ) } vEN of function elements such that: a) each is an analytic continuation of every other; b) any function element that is an analytic continuation of a function element in CAF is (also) in CAF: CAF is a maximal set of function elements each of which is an analytic continuation of any other.

• The function elements (II" ' fll" ) and (Iv , flv ) in CAF are a-equivalent, i.e. , UI" ' fll" ) "' a Uv , flv) , iff: a) a E fll" n flv ; b) in some neighborhood N(a), fl" IN (a) = fv IN (a ) " Thus to each a in the union fl �f U flv vEN there corresponds a set of "'a-equivalence classes of function elements (II" ' flv) for which a E flw

• When (I, fl) E CAF and a E fl, the "'a-equivalence class to which U, fl) belongs is the germ or a branch of (I, fl) at a, v. 10.2 .1 , and is denoted [I, a] . As the discussions of 10.1 . 10 and 10.1 . 12 reveal, it is quite possible for a function element (g, n) (for which a E n) to belong to CAF while, in the current notation, [I, a] -j. [g, a] . A result proved below and due to Poincare, implies that the cardinality of all germs at a cannot exceed No .

• The set W �f { [I, a] : a E fl, (I, fl) E CAF } of germs is , in recognition of its originator, Weierstrafi, the W-structure determined by any (hence every) function element in CAF. (A topology for W is given in 10.2.4 below.)

• Associated with W are the projection p : W '3 [I, a] r-+ a E fl and the map f : W '3 [f, a] r-+ f(a) E C.


10.2.1 Exercise. a) When a E C and P(a, z) is a power series with a positive radius of convergence Ra , then P( a, z) represents some function f and an associated germ [I, a] . b) For a fixed, the correspondence

{ P(a, z) : Ra > O } -+ { [f, a] : a E fl, f E H (fl) }

between the set Sa �f { (P( a, z) , Ra) } of all pairs consisting of a power series P( a, z) converging at and near a and the associated radius of convergence Ra and the set Hf, an of all germs at a is bijective. c) If

UI" ' fll" ) and Uy, fly ) need not be "-'b-equivalent. Nevertheless, UIL ' fllL ) and Uy, fly ) are analytic continuations of one another. d) If a -j. b two of the equations [f, a] = [g , b] , [I, a] = [g , a] , [I, a] = [I, b] , are meaningless.

For any a in fl, a function element U, fl) and one of its analytic contin-uations (g , n) such that a E fl n n, the numbers f( a) and g( a) may differ, v. 10.1.10 and 10.1 .12. Nevertheless, they are regarded as values of the multivalued function that arises from analytic continuation.

On the other hand, f is a true (single-valued) function on W and the range or image f(W) accurately reflects the different values f(a) , g(a) , . . , .

10.2.2 Exercise. The set fl '!gf U fly is a region. What is fl in 10.1 .21? "'EN

ex:> n [Hint: The series L .;. arises by integrations and algebraic transn n=! 00 formations applied to L zn .]

n=O

10.2.3 Exercise. The function f is well-defined on W. The description of CAF suggests a topology derivable by pasting to

gether the fly used to provide analytic continuations. However, the accurate description of such an informal topology is beset with the complications arising from the presence in CAF of equivalent function elements. A topology is more readily attached to the set W of equivalence classes, i.e. , germs or branches, according to

10.2.4 DEFINITION. WHEN V IS AN OPEN SUBSET OF fl AND f E H (V) ,

[I, V] �f { [f, a] : a E V } .

Section 10.2. Manifolds and Riemann Surfaces 393

10.2.5 Exercise. The set

T �f { [I, V] : V an open subset of Q, for some Q, (j, Q) E CAr }

is a Hausdorff topology for W.

10.2.6 THEOREM. WITH RESPECT TO T, p IS CONTINUOUS AND OPEN.

PROOF. If a E V E 0(((:) and [I, a] E p-l (V) , then [I, V] is a neighborhood of [I, a] and p( [J, V] ) = V: p is continuous.

For any open subset V of Q, p([I, V] ) = V, whence p is open. D

10.2 .7 Exercise. If V is an open subset of Q, then p l [J,v] is injective. (Hence p is locally a homeomorphism.)

[ 10.2.8 Note. For a in Q, a in C, and germs [I, a] , [g , a] , the germs [aJ, a] , [I + g, a] , and [lg , a] are well-defined. Thus, for each a in Q there is the C-algebra

!ia �f { [I, a] : J holomorphic in some N (a) } of germs of functions holomorphic at and near a. Generally, for a category C, e.g. , the category of C-algebras, and a topological space X, a sheaJ S [Bre] is defined by associating to each U in O(X) an object S(U) in C. It is assumed that:

• When {U, V} C O(X) and U C V, there is a morphism

Pb : S(V) r-+ S(U ) .

• When {U, V, W} c O(X) and U e V e W, then

Pw pV 0 w . u = u Pv ,

p� is the identity morphism. • When {U>.}),EA C O(X), U �f U U)" kl ' k2 in S(U ) are the

),EA same iff for each A, pg>. (kI ) = pg>. (k2 ) .

• For U), and U above, if

k), E S (U), ) , A E A,

{W �f U), n Ul" -=j:. (/)} ::::} {p� (k), ) = p� (kl" ) } ,

there is in S(U) a k such that for every >., pg>. (k) = k), .


When x E X, the filter V �f {V(x) } of open neighborhoods of x is partially ordered by inclusion: U(x) -< V(x) iff U(x) C V(x). The stalk Sx at x is the set of all {kv }v EV such that

For s E sx , if p(s) �f x, then p maps S �f U Sx onto X. The xEX

set { p- l (U) : U E O(X) } is a topology T for S; p is open and T is the weakest topology with respect to which p is locally a homeomorphism. If X is a Hausdorff space, so is S. When the objects in the category C are algebraic, e.g. , when C is the category of groups and homomorphisms or the category of C-algebras and C-homomorphisms, as x varies in X the results of the algebraic operations within the stalk Sx are assumed to depend continuously on x. For example, when X = C and C is the category of all C-algebras, for each open subset V of C, the set S(V) may be taken as the set of all functions holomorphic in V, and when U C V, pi; maps each f in S(V) into f l u ' Then:

• S is the sheaf of germs of holomorphic functions. • For a in C, Sa is the stalk at a and consists of all germs [I, a] . • The map p : S r-+ C is that given iri the context of W: p maps

each germ [I, a] onto its second component a. • A W-structure W is a connected subset of S. • The elements [oJ, a] , [I + g, a] and [lg , a] , as maps from C to

S are continuous.]

10.2.9 Exercise. The topology induced on each stalk of a sheaf is discrete. (Hence, when the stalk is an obj�ct in a category of topological objects, the (discrete) topology induced by T on the stalk need not be the same as the topology of the stalk viewed as an object in its category. )

10.2 .10 THEOREM. (Poincare) IF a E Q THE CARDINALITY OF THE SET OF GERMS [I, a] AT a DOES NOT EXCEED No .

PROOF. Each germ [I, a] corresponds to some function element (f, Qv ) (such that a E Qv) . If [g , a] corresponds to (g , QJL ) (again a E QJL) ' then (g , QJL ) is an analytic continuation of (I, Qv) . Thus there is a finite chain

in which (ik, QVk ) is an immediate analytic continuation of (ik- l , Qvk_l ) ' 2 :::; k :::; m. Each QVk contains a point p + iq for which {p, q} C Qi, i.e. ,


p + iq is a complex rational point. Thus, corresponding to each finite chain C, there is a finite sequence of complex rational points. The cardinality of the set of all finite subsets of ((f is No · D

Thus, if Z E Q and Sz �f p- 1 (z) (the stalk over z) , #(sz) :s; No . If r > 0 and V is an open subset of Q, each [I, z] (E sz ) is in a neighborhood [I, V] . Furthermore, p : [I, V] r-+ V is a homeomorphism. Consequently, W may be viewed as consisting of sheets lying above Q and locally homeomorphic to Q. If a E Q, for some positive r, V �f D(a, rt lies under a set of homeomorphic copies of V, One copy in each of the sheets. In 10.2 .11 and 10.2.12 there are precise formulations of the preceding remarks.

10.2 .11 Exercise. As z ranges over Q, the cardinality #(sz ) remams constant.

[Hint: If #(sz) �f k E N, then #(sz) = k near z. If

analytic continuation from z to w yields a contradiction.]

10.2.12 Exercise. If 1 E E, #(sz ) == 1 for the associated W-structure W.

10.2.13 Exercise. If a E Q, for some N(a) , p- 1 [N(a)] consists of (at most count ably many) pairwise disjoint homeomorphic copies of N(a) .

The following variant of the ideas above leads to greater flexibility of the discussion. The W-structures introduced thus far are extended to analytic structures that include so-called irregular points [Wey] .

00 The map I : Q 3 z r-+ L cn(z - a)n may be viewed as an analytic

n=O description of a complex curve { (z, w ) : w = � cn(z - a)n } in Q x C.

The same object may be described alternatively by the pair

00 z = a + t, W = Co + L cntn (10.2. 14)

n=1

of parametric equations . Extended somewhat further, the parametric equations (10.2.14) are replaced by a pair

00 00 P(t) �f L ant", Q(t) �f L (3rnt'n

n=k

for which the following conditions are imposed.


a) {k, l } C Z; b) P and Q converge in some nonempty punctured disc

c)

. clef D(O, rr = { t : 0 < It I < r } (if k and l are nonnegative, P and Q converge when t = 0, i .e., m D (O , rn ;

{ { {t l , t2 } C D(O, rr } 1\ {P (tI ) = P (t2 ) } 1\ {Q (tI ) = Q (t2 ) } } ::::} {tl = t2 } '

When k or l is negative, 0 is the only pole in D(O, rr of the corresponding P or Q. Hence, if, e.g . , P(O) = 00 = P (t2 ) , automatically t2 = O. Thus there is a uniquely defined injection

L : D(O, rr 3 t r-+ [P(t) , Q(t )] E C� . The parameter t is a local uniJormizer. The Junction pair (P, Q) , subjected to a )-c) , is now the object of interest.

Various parametric representations can represent the same curve, e.g.,

{ (cos t , sin t) : - 7r < t < 7r } and { ( 1 - 7: , �) : -00 < 7 < 00 } 1 + 7 1 + 7

are different parametric representations of

{ (x, y) : x2 + y2 = 1 , x > -1 } .

Function pairs [P(t) , Q(t ) ] and [R(7) , S(7)] (both subjected to a)-c)) are regarded as equivalent and one writes [P (t) , Q( t) ] rv [R( 7) , S ( 7)] iff the local uniformizer 7 is representable in the form

00 7 = L "Intn , "11 :=J 0, ( 10.2 .15)

n=l and the series (10.2 .15) converges in some open disc D(O, rr . 10.2.16 Exercise. The relation rv described above is an equivalence relation.

The rv-equivalence class containing the function pair (P, Q) is denoted 3 (P, Q) or simply 3· The set of all 3 is 3 and the elements 3 of 3 are points .


10.2 .17 Exercise. For the maps

a : 3 3 3 (P, Q) r-+ P(O) E ((:(3 : 3 3 3(P, Q) r-+ Q(O) E ((:, (a , (3) : 3 3 3 (P, Q) r-+ [P(O ) , Q(O)] E ((:2 ,

397

the complex numbers a [3 (P, Q)] and (3 [3 (P, Q)] are independent of the representative (P, Q) .

Thus the notations 0'(3) and (3(3) unambiguously define complex numbers: a and (3 are in ((:3 .

10.2.18 DEFINITION. THE TOPOLOGY T OF 3 IS THE WEAKEST TOPOLOGY WITH RESPECT TO WHICH L : 3 r-+ ((:2 DEFINED ABOVE IS CONTINUOUS. 10.2.19 Exercise. If 3 (P, Q) E 3 and D(O, rt is a nOn empty open disc on which the map L : D(O, rt 3 t r-+ [P(t ) , Q(t)] E ((:2 is injective, a typ-ical neighborhood N [3 (P, Q)] is L (D(O, rn and consists of all 3 (15, Q) described as follows.

For some to in D(O, r t and all t such that to + t E D(O, r t ,

10.2.20 Exercise. By abuse of language, N [3 (P, Q)] consists, for all immediate analytic continuations (15, Q) by rearrangement of the pair (P, Q) , of the points 3 (15, Q) . 10.2.21 Exercise. If (P, Q) rv (R, S) , each neighborhood N[3 (P, Q)] contains a neighborhood N[3 (R, S)] .

The base of neighborhoods at a point 3 in 3 may be defined without regard to the particular parametrization.

10.2 .22 Exercise. As defined by neighborhoods described above, T is a Hausdorff topology for 3.

10.2.23 Exercise. Each point 3 in 3 i s contained in a neighborhood N(3) homeomorphic to 1U.

10.2.24 Exercise. The maps a and (3 are local homeomorphisms with respect to the topology T.

10.2.25 Exercise. If [�cn (z - a)n , D(a, rt 1 is a function element,

[I, a] the corresponding equivalence class, and (P, Q) is a function pair


00 created by unijormization, i.e. , P(t) = a + t, Q(t) = L cntn , the map

n=O

I : W 3 [I, a] r-+ 3 (P, Q) E 3

is a (1", T) homeomorphism. In many discussions of sets of equivalence classes, e.g. , in LP (X, J1) ,

the distinction between an equivalence class and one of its representatives is blurred. Thus , when ambiguity is unlikely, no distinction is made between an equivalence class 3 and one of its function pairs (P, Q) . 10.2.26 Example. The following are some important examples of function pairs.

00 P(t) = a + t, Q(t) = L bmtm ,

m=O 00

m=l 00

P(t) = Ck , k E N, Q(t) = L bmtm , l E Z. m=l

The function pair in:

(10.2.27)

(10.2 .28)

( 10.2 .29)

(10.2.27) corresponds to a function that is locally invertible near a; (10.2.28) corresponds to a function that conforms to the behavior considered in 5.3.47;

(10.2.29) corresponds to a function with a pole.

10.2.30 Example. By appropriate reparametrizations, any function pair can be represented in one of the forms (10.2.27) , (10.2.28), or (10.2.29) .

(10.2.27). If the original pair is (� antn , %:;0 bmtm ) and al -j. 0, a reparametrization is

(10.2.28). If k > 1 , ak -j. 0, and the original pair is

( ao + aktk + . . . , t, bmtm ) ,


for a Ck such that c� = ak , a reparametrization is

(10.2.29 ) . If k > 0 and the original pair is

a-k :=J 0, and c� = a_k , a reparametrization is

ex:> ex:> �f ", p _ '" q _ 1 T - � cpt t - � O'qT , 0'1 - - , p=1 q=1 C1

399

Those points 3 representable in form (10.2.27) are the regular points; those representable in forms (10.2.28) and (10.2.29) are the irregular points . More particularly, points representable in form (10.2 .28) are branch points of order k; those representable in form (10.2.29) are poles of order k. When k > 1 in form (10.2.29) , the pole is branched . 10.2.31 DEFINITION. A SUBSET AS of 3 IS AN analytic structure IFF #(AS) > 1 AND : a) ANY TWO POINTS OF AS ARE THE ENDPOINTS OF A CURVE "/ : [0, 1] 3 t r-+ "/(t) E 3 ; b) WHEN 3 E 3 AND FOR SOME CURVE ,,/ , ,,/(0) E AS AND ,,/(1 ) = 3 , THEN 3 E AS.

Each W is a subset of some unique AS(W): W e AS(W) . If each point of AS(W) is a regular point, W = AS(W) . By abuse of language, an analytic structure AS is a curve-component of 3.

10.2.32 Example. Under the convention whereby � is identified with 00, o

i .e. , when the discussion is conducted in Coo , One neighborhood of the func-tion pair :F �f (P(t) = t, Q(t) = �) corresponds to { t : t E C, I t I > O } , a neighborhood of 00, v. Section 5.6. The analytic structure engendered by analytic continuation is Coo .


Owing to the generality with which function elements are defined, the analytic structure AS is larger than the analytic structure W presented earlier. The inclusion of irregular function elements permits the adjunction to W of branch points, poles, etc.

Henceforth, Weyl's adaptation [Wey] of Weierstrafi's ideas for analytic structures is invoked where it is helpful.

10.2.33 LEMMA . EACH IRREGULAR poINT 30 LIES IN A NEIGHBORHOOD N (30 ) IN WHICH ALL OTHER POINTS ARE REGULAR.

PROOF. If (P, Q) E 30, for some positive r, if ° < l a l < r, then

and p' I v¥= 0, since otherwise, the Identity Theorem implies p' == 0, a contradiction. The neighborhood N (30 ) corresponding to D(O, rt meets the requirements. D

10.2 .34 THEOREM. AT MOST COUNTABLY MANY POINTS IN AN ANALYTIC STRUCTURE ARE IRREGULAR.

PROOF. From 10.2.33 it follows that each 3 is contained in a neighborhood N (30) consisting (except possibly for 30 itself) of regular points. One of these regular points corresponds via the local homeomorphism to a p + iq in QI + iQl. Thus the irregular points are in bijective correspondence with a subset of QI + iQl (cf. PROOF of 10.2.10) . D

10.2.35 Example. If

00

clef Cn =

(�) (�) . . . [ - (2�- I) ] n! and I z - 1 1 < 1,

then L cn(z - I)n converges and represents a function fo such that for n=O

Z in D(I , It , [fo (zW = z. The two parametric representations of Vz are [R± (t) , S± (t)] � (t + 1 , � ±cntn) : there are two square roots of z.

More generally, if 2 :::; k E N, there is a sequence {cn } nE]\/+ ' found by formally differentiating zt and evaluating the results when z = 1 , and for 00 z in D(I, It , L cn(z - It converges and represents a function fo (z) such

n=O that for z in D(I , It , [Jo (zW = z: Parametrizations of (fo , D(I , 1)° ) are


[Rq (t ) , Sq (t)] �f (t + 1, � exp ( 2�7ri ) cntn) , 0 :::; q :::; k - 1 : there are k

( different) kth roots of z. Via 10.2 . 19, there are k pairwise disjoint neighborhoods

each homeomorphic to 1U. The corresponding AS provides a k-fold cover of C, v. Section 10.3.

If 0 :::; q :::; k - 1 , "I : [0, 1] 3 s r-+ "I(s) E 3 is a curve, and for s in [0, 0 .5) , "I(s) �f (Ps , Qs ) i s regular, v. (10.2.27) , while "1 (0 .5) = (Rq , Sq ) , there are k possible continuations of "1 (0 .5) along the rest "I ( [0.5, 1] ) of the curve.

If "I : [0, 1] 3 t r-+ "I(t) E 3 is a curve and, when

o :::; t :::; to < 1 ,

each "I(t) i s a regular point but "I (to) i s a branch point of order k, according to which of the k different choices of representation defines "I (to ) , the curve "I is One of k curves "11 , . . . , "Ik , say "Ii such that

if 0 :::; t < to if to :::; t :::; t 1 :::; 1 .

The phenomenon just described is the genesis of the term branch point and the particular "Ij is a branch. That "Ij may lead to another branch point for some t l in (to , 1 ) . One of the branches corresponding to a t l in (to , 1 ) can be one of the "Ii discarded at to ·

10.2.36 Exercise. In the context above, if 5 is the set of t such that "I(t) is an irregular point, #(5) E N+ .

[Hint: The argument in the PROOF of 10.2 .33 applies.] The further study of analytic structures is facilitated by the next dis

cussion of related topological questions.

10.2 .37 DEFINITION. A TOPOLOGICAL SPACE X IS : a) curve-connected IFF ANY TWO POINTS OF X ARE THE ENDPOINTS

"1 (0) AND "1 (1 ) OF A CURVE "I : [0, 1] r-+ X; b) locally curve-connected IFF FOR EACH x IN X, EACH NEIGHBORHOOD

N(x) CONTAINS A NEIGHBORHOOD V(x) SUCH THAT ANY TWO POINTS IN V(x) ARE END POINTS OF A CURVE "I SUCH THAT "1* C N(x) ;

c) simply connected IFF X IS CURVE-CONNECTED AND EACH LOOP

"I : [0, 1] r-+ X


IS LOOP HOMOTOPIC IN X TO A CONSTANT MAP; d) locally simply connected IFF X IS LOCALLY CURVE-CONNECTED AND

FOR EACH x IN X, EACH NEIGHBORHOOD N(x) CONTAINS A NEIGHBORHOOD V(X) SUCH THAT EACH LOOP "( FOR WHICH "((0) = x (THE LOOP STARTS AT x) , "((I) = x (THE LOOP ENDS AT x) , AND "(* C V(X) (THE LOOP IS CONTAINED IN V(X) ) IS LOOP HOMOTOPIC (VIA SOME CONTINUOUS MAP F OF [0, 1] 2 ) IN N(x) TO THE CONSTANT MAP 15 : [0, 1] 3 t r-+ r5(t) = x. (IN THE NOTATION OF 1 .4 .1 ,

"( "'F,N(x) 15. )

A MAXIMAL CURVE-CONNECTED SUBSET OF X IS A curve-component . 10.2.38 Example.

• The punctured open disc 10 is curve-connected, locally curve-connected, locally simply connected, but not simply connected.

• The union 1Ul:JD(2, 1r is locally curve-connected, simply connected, locally simply connected, and not connected.

• The topologist 's sine curve

{ (x, y) : y = sin (�) , ° < x � 1 } l:J { (0, y) : - 1 � y � 1 }

in the topology inherited from ffi.2 is' connected, not curve-connected,

and not locally connected (no point on { (O, y) : - 1 � y � I } lies in a connected neighborhood) .

10.2 .39 DEFINITION. A CONNECTED HAUSDORFF SPACE X IS AN ndimensional cumplex manifold IFF:

• FOR EACH POINT x IN X THERE IS A PAIR {N(x), ¢} (A chart at x) CONSISTING OF A NEIGHBORHOOD N(x) OF x AND , FOR SOME NONEMPTY OPEN SUBSET U OF en , THERE IS A HOMEOMORPHISM

¢ : N(x) 3 x r-+ U.

• FOR CHARTS {N(x) , ¢} AND {N(y) , 1jJ} SUCH THAT N(x) n N(y) -j. (/) THERE IS THE transition map

t<jJ¢ �f 1jJ 0 ¢- l : ¢[N(x)] n 1jJ(N(y) r-+ ¢[N(x)] n 1jJ(N(y) . WHEN EACH t<jJ¢ IS IN H (W) , i.e. , WHEN a �f (al , . . . , an) E W, AND THERE ARE n POWER SERIES

c(k) (Zl - al )Tnl . . , (z - a )Tnn 1 _< k _< n Tnl .. . . ,Tnn n n


CONVERGING UNIFORMLY TO t7jJ¢ ON EVERY COMPACT SUBSET OF A NONEMPTY POLYDISC X�=lD (ai , rit CONTAINED IN W, X IS AN n -dimensional complex analytic manifold . THE SET A � { {N(x), ¢} LEx IS AN atlas . Two charts {N1 (x) , ¢d and {N2 (x) ' ¢2 } are holomorphically compat

ible iff the map ¢2 0 ¢� 1 : ¢1 [N1 (x) n N2 (X)] r-+ ¢2 [N1 (x) n N2 (X)] is biholomorphic.

Two atlases �1 and �2 for a complex analytic manifold X are analytically equivalent (�1 rv �2 ) iff for each x in X, every chart {N1 (x) , ¢1 } in �1 is holomorphically compatible with every chart {N2 (x), ¢2 } in �2 . An atlas � is complete iff any chart holomorphically compatible with some chart in � is in �: � is saturated.

[ 10.2.40 Note. The assumption that X is a Hausdorff space is not redundant [Be] . For k in N+ U {oo}, there are Ck-compatible charts, and Ck_ equivalent atlases. c) The map t7jJ¢ may be regarded as a ffi.2n_ valued function on ffi.2n . clef . clef ( If Zj = X2j- 1 + �X2j and t<jJ¢ = U1 , U2 , · · · , U2n- 1 , U2n ) the sign of the determinant of the Jacobian matrix J

det (J) �f det [8 (U1 , . . . , U2n ) ] 8 (X1 , . . . , X2n )

may vary from point to point of X. When det (J) > 0 everywhere, X is oriented by the atlas and X is orientable . When there is no atlas that orients X, X is nonorientable. The Riemann sphere L2 is an orient able (I-dimensional) complex manifold. If 0 < a a } , V �f L2 n { (x, y, z) then U U V = L2 . Under the convention � = 0, 00

1 a : U 3 P

r-+ 8(P) and (3 : V 3 P r-+

8(P)

z < b } ,

provide charts {U, a} and {V, (3} that constitute an atlas for L2 .

The Mobius strip M S is the set S �f [0, 1] x (0, 1 ) reduced modulo the equivalence relation rv defined by

b d { {a = c} /\ {b = d} if a -j. O (a , ) rv (c, ) {:} {a = O} /\ {c = I } /\ {b = 1 - d} otherwise ·


More intuitively, the Mobius strip is S with O x (0, 1 ) and 1 x (0, 1 ) identified according to the rule (0 , x) == ( 1 , 1 - x) . For the maps

¢ : [0, 0.6] x (0, 1 ) 3 (x, y) r-+ (x, y) , 'ljJ : [0 .4, 1] x (0, 1 ) 3 (x, y) r-+ (1 - x, 1 - y) ,

an atlas A for M S i s the pair of charts { [O , 0.6] x (0, 1 ) , ¢} and { [0.4, 1] x (0, 1 ) , 'ljJ} . The map t<jJ viewed as a map of

W �f (0.4, 0.6) x (0, 1 ) into itself takes the form t<jJ (x + iy) = x + i (l - y) . The determinant of the corresponding Jacobian matrix is - 1 : M S is not oriented by A.]

10.2.41 Exercise. a) If Al and A2 are equivalent atlases for a manifold, the determinants of the corresponding Jacobian matrices are of the same sign. (Hence the Mobius strip is not orient able. ) b) If Al and A2 are equivalent atlases for an oriented manifold, {Nl , ¢1 } and {N2 , ¢2 } are associated charts such that Nl n N2 -j. 0, the determinant of the Jacobian matrix for h . . t clef --/" -

l ' f . t e transItIOn map <1>, <1>2 = '/-'1 0 ¢2 IS 0 constant sIgn.

[Hint: b) For charts { N1 , ¢1 } resp. { N2 , ¢2 } in Al resp. A2 such that Nl n Nl n N2 n N2 -j. 0, a calculation of the determinant of - --1 the Jacobian matrix for ¢1 0 ¢:; 1 0 ¢1 0 ¢2 applies.] The set of charts endows X with a topology and with the capacity to

support a notion of holomorphy for certain maps defined on X: X is an (n-dimensional) complex analytic manifold.

When X and Y are n-dimensional complex analytic manifolds, a function F in Y x is holomorphic near x iff for some chart {N (x), ¢} and some chart {N[F(x)] , 'ljJ} , 'ljJ o F o ¢- 1 is holomorphic on ¢[N(x)] . If F is holomorphic near each x, F is analytic.

For the special case Y = Coo , F is X -holomorph'ic iff F is analytic and F(X) c C; F is X -meromorphic iff F is analytic and F( X) c Coo . When 1l E CX , 11 is harmonic, subharmonic, superharmonic resp. har

monic, subharmonic, superharmonic near x iff for each chart resp. some chart {N ( x ) , ¢ } , 11 0 ¢- 1 E Ha{¢[N(x)] } , 11 0 ¢- 1 E SH{¢[N(x)] } , -11 0 ¢- 1 E SH {¢[N(x) ] } .

The use of charts permits many discussions of complex analytic manifolds and functions on them to be carried out locally. In particular, the


notions immediate analytic continuation and analytic continuation are formulable for appropriate function elements defined for n-dimensional complex analytic manifolds.

[ 10.2.42 Remark. A analytic structure AS (determined by some function element (I, Q)) is a special kind of complex analytic manifold . When 3 (P, Q) E AS, N[3 (P, Q)] is a neighborhood of 3 (P, Q) , and L : D(O, rr r-+ AS is the associated map, {N[3 (P, Q)] , L - I } is a chart.

Although the study of complex analytic manifolds is a large and important field of mathematics, only the definition of a complex analytic manifold is offered here. Useful references are [BiC, Nar, SiT, St] . ]

10.2 .43 DEFINITION. A Riemann surface IS A I-DIMENSIONAL COMPLEX ANALYTIC MANIFOLD .

In what follows, most of the discussion will deal with Riemann surfaces. The W-structures W and analytic structures AS have served the purpose of motivating the treatment. 10.2.44 Exercise. If R is a Riemann surface determined by an atlas A, there is a holomorphically compatible atlas Al such that each chart {N(x) , ¢} of Al is canonical, i .e. , ¢[N(x)] = 1U and ¢(x) = o. (The atlas Al is a canonical atlas. )

[Hint: Riemann's Mapping Theorem 8 . 1 . 1 applies. ]

10.2.45 Exercise. a) If W is a W-structure and [J, V] E T, then { [J, V] , p} is a chart. With respect to the set of all such charts, the W -structure determined by the function element (I, Q) is a Riemann surface. 10.2 .46 Exercise. A Riemann surface is orientable.

[Hint: A transition function t'j;.p is a (vector) function ( � ) m

(lie t2 The Jacobian matrix .J for t<jJ.p is the 2 x 2 matrix ( Ux vx ) . uy Vy

The Cauchy-Riemann equations apply, v. 8.1 .6-8 .1 .7.] 00

10.2.47 Exercise. For each convergent series L cn(z - a)n and its circle 'n=O

of convergence D( a, r r , the formul32 = =

f(z) �f L cn(z - a)n , P(t) �f a + t, and Q(t) �f L cntn n=O 71=0


establish a correspondence between the germ [I, a] and the equivalence class 3 (P, Q). Then I : W 3 [I, a] r-+ 3 (P, Q) E 3 is an injective and holomorphic map of the Riemann surface W into the Riemann surface AS, the Riemann surface determined by 3 (P, Q) (and contained in 3) . 10.2.48 Example. The open unit disc 1U i s a Riemann surface. If f E H (1U) and 8(1U) is a natural boundary for f, then 1U is the analytic structure determined by the function element (I, 1U) . Hence, if (g , Q) is an immediate analytic continuation of (I, 1U) , then Q c 1U. Thus, if a E Q, any neighborhood of the germ [g, a] is a subset of the neighborhood [I, 1U] .

10.2.49 Example. The complex plane C is a Riemann surface. If f is entire, C is the analytic structure determined by the function element (f, q . One atlas consists of the single chart {C, id }. 10.2.50 Example. The set Coo viewed as the one-point compactification of C, is a Riemann surface. An atlas for Coo is given in 10.2.40. The function element (id , Coo ) determines Coo . The function id is holomorphic in the sense that it is analytic on Coo and id (Coo ) = Coo . Liouville' s Theorem (5.3.29) implies that H (Coo ) consists entirely of constant functions.

In 10.3.21 and 10.3.23 there are precise formulations of the U niformization Theorem, a significant generalization for simply connected Riemann surfaces of 8 .1 . 1 . Two of the closely related consequences, 10.3.24 and 10.3.26 are discussed as well. In Sections 10.5 and 10.6 there are sketches of the proofs of the U niformization Theorem.

10.2.51 Exercise. The Riemann surface R corresponding to 10.1.6 is homeomorphic to C.

[Hint: If Sn �f { z : z = reiO , 0 < r < 00, -mr < e � mr } and S �f { awl + (3W2 : (a, (3) E [0, 1 )2 } , then R is homeomorphic to: a) U Sn resp. b) S.]

nEZ

10.2 .52 Exercise. A Riemann surface is locally compact and curveconnected, locally curve-connected, and locally simply connected.

[Hint: For each chart {N (x) , ¢ } , ¢ is a homeomorphism.]

10.2.53 DEFINITION. FOR AN ANALYTIC STRUCTURE AS DETERMINED BY A FUNCTION ELEMENT (I, Q) , AN OPEN SUBSET OF U OF AS, SUCH clef I THAT P = P u IS INJECTIVE IS A patch . 10.2.54 Exercise. For a region 11 of AS, a function F in CQ is holomorphic on 11, i.e. , F E H (11) , iff for each patch U contained in 11,

F 0 (p- l ) E H[P(U) ] .


10.2.55 THEOREM. IF AS IS DETERMINED BY THE FUNCTION ELEMENT (I, Q) THE CORRESPONDING FUNCTION f IS HOLOMORPHIC ON AS. PROOF. If U is a patch, some [I, a] is in U and p is injective on U. Furthermore p(U) �f V is an open subset of C and f 0 p- l = I l v . D

There is a parallel between the study of functions 1 holomorphic in a region Q of C and functions F : AS r-+ C holomorphic in a region 11 on the analytic structure AS associated to I.

A local uniformizer t maps a neighbor hood N (a) into C. 10.2.56 Exercise. A function F : AS r-+ C is holomorphic in a region 11 iff for each 30 in 11 there is a local uniformizer t such that in some neighborhood of 30 , F is representable by a convergent power series:

00

n=O

[ 10.2 .57 Remark. If Q c C, a function I : Q 3 z r-+ I(z) E C of one complex variable gives rise to an equation w - I( z) = 0 involving two complex variables. The function 1 in 10.1.6 satisfies ef(z) - z = 0 and the function g in 10. 1 . 12 satisfies g (z)2 - z = o. Each equation is an instance, for some subset £ of C2 and a function F : £ 3 (w, z) r-+ F( w , z) E C, of an equation of the form F(w , z) = 0 in which w is replaced by I(z) resp. g(z) . Just as

{ (x, y) : ( x, y) E ffi? , x = (1 _ y2) ! } � { (x, y) : (x, y) E ffi? , x2 + y2 - 1 = 0 } ,

the inclusion { (w, z) : w - I (z) = o } � { (w, z) : F (w, z) = o }

when w is replaced by 1 can obtain. Both sides of the latter inclusion describe parts of a CAY, hence of a W, hence of an R. In the cited illustrations, if (a, b) E £, F is representable at and

00 near (a, b) by a power series L CTnn (w - a)Tnm(z - b)n . Since

'TTl ,n=O the series converges absolutely near (a, b) ,

00

rTL,n=O

00 clef ", ) ( )n = � Cr' (w z - b . n=O


00 If C 1 ( a) -j. 0, there is a power series L dk (w - a) k such that near

k=O

a, F ( w, � dk (w - a)k) = 0 (v. 5.3.43) .

If F(w, z) �f w - z2 and a = b = 1 , then C1 (1 ) = -2 and when w is near 1 there are two power series

Z = ±2 (l + � (W - l) + m (�) (W - l)' + ) �f ±P(I , w) 2 21 such that F[w , ±P(I , w)] = o. The continuing (P, D(I, lr) along the curve "I : [0, 1] 3 t r-+ e21Tit results in (-P, D(I , lr ) . The same CAr is engendered by (P, D(I , In and (-P, D(I , lr ) . On the other hand, if F(w, z ) �f w2 - z2 (cf. 10 .1 .33) , then

C1 (1 ) = 2 and z = ±[(w - 1 ) + 1] [�f ±P(I , w) = ±w] . In this instance, both P and -P are entire and analytic continuation along "I : [0, 1] 3 t r-+ e21Tit of either of (P, D(I , lr) or (-P, D(I , lt ) does not result in the other bnt ouly in the function element with which the continuation was begun. The CAY associated with (P, D(I , lr) differs from that associated with (-P, D(I , ln· 00 When F(w, z) = L cmn (w - a)m (z - bt, a series converging

rn.n=O at and near (a , b) , as a subset of ([2 , { (w, z) : F(w , z) = O } may be regarded as the graph Qp of the equation F(w, z) = o. The topology of ([2 induces a topology on Qp . When (I, Q) is a function element such that for w in Q, F[w, f(w)] = 0, for each w in Q, [w , f(w)] E Qp . The CAY associated with (I, Q) corresponds to a (possibly proper) subset of QF. Consequently a CAY, a W, and more generally an R may be viewed as a (possibly proper) subset of some Qp . Owing to the method of topologizing a CAY, a W, or an R, these subsets, like Qp itself, are for the most part locally conforrnally equivalent to 1U. However, as the consideration of the case F( w, z) �f w2 - z2 reveals , Q F and one of the corresponding CAY, W) or R are not necessarily homeomorphic. The global topological character of CAY and W is determined by the process of analytic continuation whereby any two neighborhoods may be connected. When analytic continuation is attempted at

Section 10.3. Covering Spaces and Lifts

some neighborhood of OF , not every other neighborhood of OF is necessarily accessible. For example, the graph OF of F(w, z) = w2 - z2 is a connected set, namely the union of two complex stra'ight lines intersecting at (0, 0) . Nevertheless their equations, w = ±z, engender two (different! ) complete analytic functions CAF± that lie in disjoint components of the sheaf S, v. 10.2.8. ]

10 .3 . Covering Spaces and Lifts

409

The triple (W, Q, p) consisting of a W-structure W and its associated region Q resp. local homeomorphism p : W r-+ Q is an example of a covering space triple conforming to the general pattern (X, Y, p) described below.

The contents of 10.2 . 18 - 10.2.25, treating the topological properties of a Riemann surface, are central to the discussion.

To simplify the presentation, unless the contrary is stated, each topological space introduced below is assumed to be a curve-connected, locally connected (whence locally curve-connected) , and locally simply connected Hausdorff space.

10.3.1 DEFINITION. A covering space tr'iple (X, Y, p) CONSISTS OF TOPOLOGICAL SPACES X AND Y AND A MAP p : X r-+ Y SUCH THAT: a) p IS A CONTINUOUS SURJECTION; b) FOR EACH Y IN Y AND SOME OPEN NEIGHBORHOOD N(y) , p- l [N (y) ]

IS THE UNION OF PAIRWISE DISJOINT X-OPEN SUBSETS , EACH HOMEOMORPHIC TO N(y) (N(y) IS evenly covered ) . FOR A CURVE "I : [0 , 1] r-+ Y AND A POINT A IN p- l b(O)] , A lift of "I

through A IS A CURVE ;:y : [0 , 1 ] r-+ X SUCH THAT ;:y(0) = A AND p o ;:Y = "I .

[ 10.3.2 Remark. Conventionally, X is a covering space of Y. The DEFINITION above is most convenient for the purposes below. Alternative definitions, some more and some less restrictive, can be found in the extensive literature of topology.]

10.3.3 Example. a) As remarked above, a W-structure W is a covering space of the underlying region Q.

b) For any topological space X, (X, X, id ) is a covering space triple. c) For

X (_�f (0, 1 )2 , Y d_�f (0, 1 ) , d X ( b) Y an p : 3 a, r-+ a E ,


P - 1 ( a) = a x (0, 1 ) , which is not homeomorphic to a union of pairwise disjoint X-open subsets homeomorphic to a: (X, Y, p) is not a covering space triple. 10.3.4 Exercise. If X is a covering space of Y and Y is a covering space of Z, then X is a covering space of Z. 10.3.5 Exercise. If X is a covering space of Y and y E Y, for some N(y) , each component of p- l [N(y)] is homeomorphic to N(y) . 10.3.6 THEOREM. FOR A COVERING SPACE TRIPLE (X, Y, p) AND A CURVE "I : [0, 1] f-t Y SUCH THAT "1(0) �f a, IF A E p- l (a) , THERE IS A UNIQUE LIFT ;:y : [0, 1] f-t X of "I THROUGH A.

[ 10.3 .7 Note. The result 10.3.6 is central to much of what follows, in particular to 10.3.12 .]

PROOF. Existence of ;:Y. Some neighborhood N(A) is open and curveconnected, and p I N(A ) i s a homeomorphism. Hence p[N(A)] �f N(a) is open and curve-connected. Consequently, for some t l in (0, 1] if O :::; s < t l , then "I(s) E N(a) . Thus, if "Is (t) �f "I(st) , ¢o : N(a) f-t N(A) is the N(a)local inverse of p and s < t 1 , then ;:y s �f ¢o 0 "Is is a lift of "Is through A. If S is the (nonempty) set of s for which there is a lift ;:Ys of "Is through A, then [0, tI l C S, S is connected, sup S �f a > 0, and S = [0, a] .

If a < 1 , some curve-connected neighborhood N b,,(l) ] is homeomorphic, say via ¢,,' an N b,,(l)]-local inverse of p, to a curve-connected neighborhood N [;:y,,(1 )] . For some ( in (a, 1] ,

{a < s < (} '* hs (l) E N b,, (l) ] } , and ¢" 0 "Ie, is a lift of "Ie, through A. Thus a i- sup S, a contradiction: a = l .

Uniqueness of ;:Y. If i] is a second lift of "I through A and

then ° E T i- 0, and continuity considerations imply T is closed. If t E T, some N [i](t)] (= N [;:Y(t) ] ) is p-homeomorphic to a neighborhood

N [1] (t)] (= N {p[i](t) ] } ) . Some neighborhood N(t) is mapped by "I into N[1J(t)] . If r E N(t ) , then p [i](r)] = p [;:Y(r)] . Since p I NF(t)] is a homeomorphism, it follows that r E T: T is open. Because [0, 1] is connected, T = [0, 1] . D

Section 10.3. Covering Spaces and Lifts 411

In the following paragraphs there is described the construction of the universal covering space X for a topological space Y.

For a fixed point ao in Y and a (variable) point a in Y, the set Sa of all curves "I : [0, 1] r-+ Y for which "1(0) = ao and "1(1 ) = a is decomposable into equivalence classes {Sa,OI } OlEA for the following equivalence relation.

Curves "I and 15 are equivalent ("I eva 15) iff: • "1(0) = 15(0) = ao (both curves start at ao) ; • "1(1 ) = 15(1) = a (both curves end at a) ; • for some continuous F : [0, 1] 2 3 {x, t} r-+ Y,

F(x, O) = ao, F(x, 1) = a, F(O, t) = "I(t) , F(l , t) = r5(t ) , t E [0, 1] ,

("I and 15 are homotopic) . clef There emerges the set X = {Sap } OlEA '

An element A of X is determined by: • a point a of Y; • a curve "I such that "1(0) = ao and "1(1) = a. (The homotopy

equivalence class of "I is A.)

If A E X, there is in Y a (unique) a such that some homotopy equivalence class of curves starting at ao and ending at a engenders A. Hence there is defined a map p : X 3 A r-+ a E Y. The set X is given the topology T for which a neighborhood base is the totality of all neighborhoods as described next.

For a point A �f Sa,OI in X, a "l in A, and a neighborhood N(a) , a neighborhood N(A) consists of the union, taken over all b in N( a) , of the set of all homotopy equivalence classes of curves that are products "115 of "I and curves 15 such that 15(0) = a, 15(1) = b, and 15* c N(a) . In brief, each point a of Y gives rise to a set of homotopy equivalence

classes. For each point b in some neighborhood N (a) of a, there is a set Sb of homotopy equivalence classes, each consisting of all pairwise homotopic curves for which the curve-images start at ao , pass through a, thereafter remain in N(a ) , and end at b. The neighborhood of A is

N(A) �f U Sb . bEN(a)

Thus a neighborhood N(A) of an A in X is determined by:

• a curve "I such that "1(0) = ao and "1(1 ) = a �f p(A) ;


• a neighborhood N(a) . The set X as described above is the universal covering space for the

space Y. 10.3.8 Example. If Y = 'lI', then Y is not simply connected. The infinite helix X � { (x, y, z) : x = cos 27rt, Y = sin 27rt, z = t, -00 < t < 00 } is the universal covering space of 'lI'. For

p : X 3 (cos 27rt, sin 27rt, t) r-+ cos 27rt + i sin 27rt E 'lI',

(X, 'lI', p) is the covering space triple.

10.3 .9 THEOREM. a) THE UNIVERSAL COVERING SPACE X IS A HAUSDORFF SPACE; b) (X, Y, p) IS A COVERING SPACE TRIPLE; c) X IS CURVECONNECTED , LOCALLY CONNECTED (HENCE ALSO LOCALLY CURVE-CONNECTED ) , AND LOCALLY SIMPLY CONNECTED . PROOF.

a) If W and Z are two points in X and w �f p(W) -I p(Z) �f z, there exist disjoint neighbor hoods N ( w) and N ( z) . Hence the corresponding neighborhoods N(W) and N(Z) are disjoint. def On the other hand, if w = p(W) = p(Z) , there are curves 'Yw and 'YZ connecting ao and w and such that:

'YW : [0, 1] r-+ Y, 'Yz : [0, 1] r-+ Y, 'Yw (O) = 'Yz(O) = ao , 'Yw ( l ) = 'Yz(l ) = 11) ,

'YW is not homotopic to 'YZ .

Moreover, w lies in an open, curve-connected, and simply connected neighborhood N(w) . The pairs bw , N(w)] and bz , N(w)] determine neighborhoods V(W) and V(Z) of W and Z. If E E V(W) n V(Z), there are curves I' : [0, 1] r-+ Y resp. 15 : [0, 1] r-+ N(w) for which

1'(0) = ao , 1'(1 ) = w resp. 15(0) = w, 15(1 ) = p(E) and such that 'YE �f 1'15 determines E. Since E E V(W) n V(Z), the homotopy equivalence classes bE}, bwr5}, and bzr5} are the same, i.e. ,

br5} = bwr5} = bzr5} , bw } = bz} ,

a contradiction, since 'YW and 'YZ are not homotopic. The remaining axioms for a topological space are directly verifiable,

particularly in light of 1 . 7 .2 . b) If A E X and p(A) = a, any neighborhood N(A) is mapped by p

onto some neighborhood of a: p is open.


If A E X, p(A) = a, and N(a) is a neighborhood in X, N(a) may be assumed to be simply connected. Then N (a) and some curve "( such that "((0) = ao and "((I ) = a determines a neighborhood N(A) and, by definition, p[N(A)] = N(a) : p is continuous.

If A E X and p(A) = a, because Y is locally simply connected, for some curve-connected open neighborhood N(a) , any loop containing a and contained in N(a) is null homotopic in Y. The neighborhood N(a) and some curve "( such that ,,((0) = ao and "((I) = a determines a neighborhood N(A). If W and Z in N(A) are determined by curves "(8 and "(( and if p(W) = p(Z) �f b [E N(a)] , then "(8 and "(( are homotopic, and thus W = Z: p is injective.

As an injective, continuous, and open map, p I N(A) is a homeomorphism.

If A -I W E p- l (a) , N(W) is the associated N(a)-induced neighbor-hood of W, and K E N(W) n N(A) , then p(K) �f C E N(a) . If the curves that determine K, W, A are resp. "((, "(1], ,,(e, then "(( and "(1] are homotopic, "(( and "(e are homotopic, whence "(1] and "(e are homotopic. Thus A = W, a contradiction. Thus different N (a ) -induced neighborhoods are disjoint: p - 1 [N (a)] is the union of pairwise disjoint homeomorphic copies of N ( a) .

c) If 1]s : [0, 1] 3 t f-t Y is a set of curves, each starting at ao and depending continuously on the pair (s , t ) , for each s, 1]s determines a point As in X: p (As ) = 1]s ( l ) . If N (As ) is a neighborhood of As , there is some curve-connected open neighborhood N [1]8 (1) ] to which is associated a neighborhood U (As ) contained in N (As ) . The map p I N(As ) �f as IS a homeomorphism. If I hl is small, 1]s+h ( l) E N [1Js ( l ) ] , whence

As depends continuously on s . In particular, if "( determines some A in X, i.e. , if ,,((0) = ao , "((I) = a,

then "(s : [0, 1] 3 t f-t ,,((st) E Y depends continuously on the pair (s , t) and thus As depends continuously on s. The curve ;:Y : [0 , 1] 3 s f-t As E X connects A to the point Ao corresponding to the loop f£ : [0 , 1] f-t ao that starts and remains at ao : X is curve-connected.

Since p is a local homeomorphism, the local properties of Y are local properties of X : X is locally connected and locally simply connected. D 10.3 .10 Exercise. The universal covering space X, modulo homeomorphisms, does not depend on the choice of ao .

[ 10.3 . 11 Note. For a given curve-connected space Y and its universal covering space X, there is an action, described next, of the group 7rl (Y) (v. 5.5.4) on the universal covering space X. Although X does not depend on the choice of ao and 7rl (Y, Yo ) does


not depend on the choice of Yo , the discussion below is phrased in terms of some ao and a particular Yo .

For ao fixed in Y and A in X, there i s a curve "I starting at ao , ending at "1(1) �f a, and engendering the homotopy equivalence class that is the element A of X. If p E 7rdY, "1(1 ) ] , then p is represented by a loop 1] starting and ending at "1(1 ) . The product 1]"1 is a curve starting at ao and ending at "1 ( 1 ) . The homotopy equivalence class A of 1]"1 is the result of the action of p on A: A = p . A. Since the constructs employed are independent of the choices ao and "1(1 ) , P . A is well-defined in terms of p in 7rl (Y) and A in X.

For A fixed in X, the set O(A) �f { p . A : p E 7rl (Y) } is the 7rl (Y) orbit of A. The projection p : A r-+ a carries each point of the orbit of A into a. The set 0 of orbits and Y are in bijective correspondence via p. The customary notation for this situation is Y rv X/7rl (Y) . ]

Paraphrased, the next result asserts that lifts of homotopic curves are homotopic and that their starting and ending points are the same. It is not only intuitively appealing but is the basis for a number of important conclusions, e.g., 10.3 . 14-10.3 . 17.

10.3 .12 LEMMA. IF a) (X, Y, p) IS A COVERING SPACE TRIPLE; b)

Ao E X, p (Ao ) = ao ;

c) F : [0, 1] 2 3 (s , t) r-+ "Is (t) E Y IS CONTINUOUS ON [0, 1] 2 , i.e. , GENERATES A HOMOTOPY BETWEEN "10 AND "11 ; AND d) "Is(O) == ao , "18 (1) == b; THE LIFTS ;;;0 AND ;;;1 THROUGH Ao ARE HOMOTOPIC, AND END AT THE SAME POINT: ;;;0 ( 1 ) = ;;;1 ( 1 ) �f B.

PROOF. Each "Is has a unique lift ;;;s through Ao. Naturally associated to F is F : [0, 1]2 3 (s , t) r-+ ;;;s (t) E X. Because X and Y are locally homeomorphic, the technique used in the PROOF of 10.3.6 applies: F is continuous and the curves ;;;0 and ;;;1 are homotopic.

By definition, the connected sets ;;;o ( [0 , 1] ) resp. ;;;1 ( [0 , 1] ) are contained in the totally disconnected sets p- l (ao ) resp. p- l (b) . Thus ;;;o ( [0 , 1] ) = Ao resp. ;;;1 ( [0 , 1 ] ) is a point: ;;;0 (0) = ;;;1 (0) = Ao and ;;;0 ( 1 ) = ;;;1 ( 1 ) = B.

D

10.3.13 THEOREM. EACH UNIVERSAL COVERING SPACE IS SIMPLY CONNECTED (cf. 10.2.37c) ) .


PROOF. For a point p in Y there is the homotopy equivalence class P (in X) of loops beginning and ending at p and loop homotopic to the constant curve l1 : [0 , 1] 3 t r-+ l1(t) == p. If ;::; : [0, 1] 3 7 r-+ ;::;(7) E X is a loop beginning and ending at P, for each 7, p o ;::;(7) �f 1' (7) E Y and 1' (0) = p o ;::; (0) = p(P) = p. For each s in [0, 1 ] , the curve

;::;s : [0, 1] 3 7 r-+ ;::;(S7) E X

is a lift through P-by virtue of 10.3.6, the unique lift through P-of the curve I's : [0, 1] 3 7 r-+ I'(S7) E Y, and I's (O) = p. If ;::;(s ) �f Qs , then I's (l ) �f qs = p o ;::;(s)p (Qs ) , i .e. , Qs is the homotopy equivalence class of the curve I's , in particular, P, which is the homotopy equivalence class of r5 �f p 0 l1 is also the homotopy equivalence class of 1'1 (= 1') : ;::; = l1.

D The next results show in what sense the universal covering space X of

a space Y is universal.

10.3.14 THEOREM. IF Y IS SIMPLY CONNECTED AND (Z, Y, a) IS A COVERING SPACE TRIPLE, Z AND Y ARE HOMEOMORPHIC. PROOF. By definition, a is continuous, open, and surjective. The next lines show that a is also injective (whence a homeomorphism).

If {p, q} c Z, there is a curve r : [0, 1] r-+ Z such that r(O) = p and r(l ) = q. If a(p) = a (q) �f y, then a 0 r �f l' is a loop, and, since Y is simply connected, l' is null homotopic. The (unique) lift of l' through p is r, and the unique lift through p of the constant map r5 : [0 , 1] r-+ a (p) is l1 : [0, 1] r-+ p. Thus 10.3.12 implies p = q. D

10.3. 15 THEOREM. IF X IS THE UNIVERSAL COVERING SPACE OF Y AND (Z, Y, a) IS A COVERING SPACE TRIPLE SUCH THAT Z IS SIMPLY CONNECTED, Z AND X ARE HOMEOMORPHIC. PROOF. Each homotopy equivalence class in Z is mapped by a into a homotopy equivalence class in Y and thus to an element of X. Thus, W denoting the universal covering space of Z, there is a map F : W r-+ X.

If: a) Zo i s the basis of the construction of W; b) o· (zo) �f Yo ; and c) Yo is the basis of the construction of the universal covering space X, 10.3. 12 implies, by abuse of language, that each homotopy class in Y lifts to a unique homotopy class (through zo) of Z: there is a map G : X r-+ W. Direct examination of the maps reveals that F o G and G 0 F are the identity maps. By the same token, both F and G are continuous: W and X are homeomorphic. 10.3.14 implies that W and Z are homeomorphic. D

10.3.16 Exercise. The universal covering space X of a space Y may be characterized as the simply connected covering space of Y.


10.3.17 Exercise. For a given space Y, the set C of all covering spaces is a poset with respect to the order : Xl >- X2 iff for some P12 , (Xl , X2 , P12) is a covering space triple. Relative to >- , C has a maximal element. This maximal element is the universal covering space of Y.

The results 10.3.6, 10.3.12 , and 10.3.15 apply when Y is an analytic structure AS (with its attendant set of function elements and region Q) and X is its universal covering space S. More generally, the same results apply when Y is a Riemann surface and X is its universal covering space. In the S-AS context, a denoting the projection of S onto AS, there are two covering space triples, (AS , Q, p) and (S, AS, a) .

10.3.18 Exercise. If Y is a Riemann surface with its attendant atlas A and X is the universal covering space of Y, for X there is an atlas B such that with respect to B : a) X is a Riemann surface; b) the map p : X r-+ Y is locally biholomorphic.

[Hint: a) If {N(y) , ¢} is a chart at y, p(x) = y, and p IN(x) is a homeomorphism, {N(x) , ¢ 0 p} �f {N (x) , a} can serve as a chart at x. b) If {Nl (x) , ;3} is a second such chart at x, then

a 0 ;3- 1 E H {a (N(x) n ;3 [Nl (x)J ) } .J

10.3.19 Exercise. Any two simply connected covering spaces of AS are holomorphically equivalent.

An important consequence of the simple connectedness of S is a considerable generalization of Riemann's Mapping Theorem. The latter may be stated as follows.

If Q �e and Q is simply connected, Q is holomorphically

equivalent to 1U. From this point forward, the context is a Riemann surface R with its

attendant complete atlas � defined via canonical charts { {Va, ¢n } aEA } ' i .e. , charts such that ¢n (Vn ) = 1U.

When 0 < r < 1 and {Va , ¢a} is a chart, ¢-;; l (D(O, rt) a conformal disc, denoted { z : I z l < r } . By extension, the complement of the conformal disc { z : Iz l < r } is { z : I z l 2': r } �f R \ { z : I z l < r } and a punctured conformal disc is { z : 0 < I z l < 1 } �f Vn . In the spirit of the same convention, for u in en and 3 in R, u (z ) denotes u(3) . As clarity requires and circumstances permit, both kinds of notations are used in what follows.

Section 10.3. Covering Spaces and Lifts

TRUE FALSE FALSE

FALSE TRUE FALSE

Table 10.3.1

10.3.20 DEFINITION. A RIEMANN SURFACE R IS;

TYPE

hyperbolic elliptic

parabolic

417

• hyperbolic WHEN 5) ; FOR EACH a IN R, THE GREEN'S FUNCTION AT a, G ; R \ {a} 3 z f-t - In I z - a l EXISTS;

• elliptic WHEN IE; R IS COMPACT; • parabolic WHEN ,5) 1\ ,IE; R IS NEITHER HYPERBOLIC NOR ELLIPTIC.

Table 10.3. 1 shows how the adjectives are paired with the paradigms 1U, Coo , and C. More details of the pairings are given in 10.6.18 . In the course of the remainder of the Chapter, the distinguishing features noted in 10.3.20 and valid for the simply connected Riemann surfaces 1U, Coo , and C are shown to characterize, modulo biholomorphic equivalence, all simply connected Riemann surfaces.

10.3.21 THEOREM. IF R IS A SIMPLY CONNECTED RIEMANN SURFACE, M DENOTING ONE OF 1U, C, OR COO , THERE IS A BIHOLOMORPHIC MAP H ; R f-t M .

[ 10.3.22 Note. The result above i s variously called the Uniformization Theorem or the (General) Riemann Mapping Theorem. Owing to the fact that no two of 1U, C, and Coo are holomorphically equivalent, the M in 10.3.21 is uniquely determined by the adjective-hyperbolic, parabolic, or ellipticdescribing R. For any Riemann surface R and its universal covering space R there is the group 9 of cover transformations (Germ�n; �ecktransformationen)-biholomorphic automorphisms g ; R --+ R for which P(g(3t= P(3) · A function f holomorphic on R can, by a kind of 9-averaging such as that used in 8.4.14, be converted to a function f that is 9-invariant; f is defined on R.


In view of 10.3 .21 , a function holomorphic on one of the three paradigms-V, C, CcJO -is converted to a function on the underlying R. Owing to Liouville's Theorem, there is no nonconstant holomorphic function on Coo . There are however nonconstant meromorphic functions on Coo . Averaging these via 9 provides nonconstant meromorphic functions on R.]

Among the important consequences of the Uniformization Theorem is the following, given here without proof.

10.3.23 THEOREM. A NONCOMPACT RIEMANN SURFACE IS THE ANALYTIC STRUCTURE FOR SOME NONCONSTANT FUNCTION ELEMENT (I, Q) . A COMPACT RIEMANN SURFACE IS THE ANALYTIC STRUCTURE FOR A NONCONSTANT MEROMORPH IC FUNCTION.

Some regard 10.3.23 as the Uniformization Theorem, since behind it is the following important assertion also given here without proof.

10.3.24 THEOREM. IF R IS A RIEMANN SURFACE, R IS ITS UNIVERSAL COVERlNG� SPACE, AND 9 IS THE GROUP OF COVER TRANSFORMATIONS, THE SET RI9 OF 9-0RBITS , ENDOWED WITH THE CANONICAL QUOTI�NT TOPOLOGY, ADMITS A COMPLEX ANALYTIC STRUCTURE WHEREBY RI9 AND R ARE HOLOMORPHICALLY EQUIVALENT.

By definition, every complex analytic manifold R is first countable , i .e. , at each point x of R there is a countable set {Nn(x) } nEN of neighborhoods of x so that if N(x) is any neighborhood of x, for some no, Nno (x) C N(x) .

10.3.25 Exercise. A Riemann surface is second countable: there is a countable set {Un(x) }nEN such that if U E O(R) , then U = U Un. (The

Un C U second countability of a complex manifold is not implicit in its definition [Be] . )

[Hint: 10.3.21 applies.] [ 10.3.26 Note. The second countability of a Riemann surface R can be derived without using the Uniformization Theorem. In [Nev2] differentials, i.e., I-forms, are used to endow R with a metric (cf. 10.5.3) and, via Schwarz's alternating process and Harnack's Theorem (6.2.24) , to show that R contains a countable dense subset . Yet another approach to the second countability of R is found in 10.6.9.

In [Nev2] ' one of the ingredients in the proof of the Uniformization Theorem is the second countability of R.]

Section 10.5 is devoted to a sketch of the principal results leading or related to the Uniformization Theorem 10.3.21 .

Section lOA. Riemann Surfaces and Analysis 419

10.4. Riemann Surfaces and Analysis

Analytic continuation of a function element can lead to the phenomenon exhibited in 10. 1. 10. On the other hand, for f and Q as in 10.1 .6, if (fJ ,�

Qj ) , 1 :s: j � � (gk ' nk) 1 ::;; k :s: K, are chains, and II = gl = f, Q1 = Ql , and QJ = Qk , then fJ = gK ·

What lies at the basis of the conclusion above is the simple connectedness of Q. The general phenomenon, monodromy, is detailed in the next paragraphs.

When "/ : [0 , 1] f-t C is a curve, {(lj , Qj ) } 1 < .<J is a chain of function _ J _ J elements such that U Qj ::) ,,/* , ,,/ (0) E Ql , and ,,/ (1 ) E QJ, the function

j= 1 element (h , QI ) is an analytic continuation along the curve "/ .

10.4.1 Example. For the function f of 10.1 .23, if

"/* C 1U, ,,/ (0) = 0, and 0 E Q c 1U,

there is an analytic continuation of (I, Q) along T On the other hand, if J(t) = 2t, t E [0 , 1] ' since J* rt. lU, there is no analytic continuation of (I, Q) along J.

10.4.2 LEMMA. IF: a) (I, QI ) IS A FUNCTION ELEMENT; b) W AND Q HAVE THE MEANINGS ASSOCIATED WITH SUCH A FUNCTION ELEMENT; c)

) clef "/ IS A CURVE SUCH THAT ,,/ (0 = a E Ql ; AN ANALYTIC CONTINUATION OF (I, Ql ) ALONG "/ EXISTS IFF "/* C Q. PROOF. If there is an analytic continuation, e.g. , corresponding to the chain { (fJ , Qj ) } l«'j«'J ' of (I, QI ) along ,,/, then "/* C Q.

Conversely, if "/* C Q and a �f [J, a] , through a, there is a unique lift ;Y of "/. For each ;Y(t) in ;yo there is a neighborhood N [;Y(t)] on which p is a homeomorphism. Because ;Y* is compact, there are ti such that J o �f to < t l < . . . < tJ �f 1 and ;Y* C U N [;Y (tj )] . If

j= 1

p {N [;Y (tj )] } �f Qj , 1 :s: j :s: J, p (t j ) �f Tj , 1 :s: j :s: J,

[Jj , Tj] = ;Y (tj ) , then { (lj , Qj ) } 1 < . < J is a chain yielding an analytic continuation along "/ .

_J _ D


10.4.3 Exercise. If { (Ii, Qj ) } I « J and { (gk , Qk) } are chains _J_ l <k<K

along a curve "I, i.e. , "1* C (U Qj) n (U Qk) , and II I:n�, = gl l n,nn" J=1 k=1 then fJ ln]nnK = gK l nJnnK •

[ 10.4.4 Note. The result in 10.4.3 may be paraphrased as follows. There is at most one analytic continuation along a curve.]

The conclusion in 10.3.12 lies at the heart of the next result. In broad terms it says:

All analytic continuations along curves that start at some point a and end at some point b are the same if the curves are pairwise homotopic in a region that admits analytic continuation along all such curves contained in the region.

10.4.5 THEOREM. (Monodromy Theorem) FOR A W-STRUCTURE W AND ITS ASSOCIATED REGION Q (cf. 10.2 .2 ) , IF: a) "Ii : [0, 1] f-t Q, i = 1 , 2 ,

) clef ARE CURVES, EACH STARTING AT "11 (0) = "12 (0 = a (E Q) AND ENDING AT "11 (1 ) = "12 (1) �f b; b) FOR SOME CONTINUOUS MAP F : [0 , 1] 2 f-t Q, "11 "'F,n "12 hI AND "12 ARE HOMOTOPIC (VIA F) IN Q) ; c)

ARE CHAINS ENGENDERING ANALYTIC CONTINUATIONS

J K ALONG "11 resp. "12 AND U Qj U U r2k C Q; d)

j= 1 k= 1

a E (G1 n r21 ) , b E (G J n r2K) , g1 1 G1 nnl = h1 I G,nn, ;

ANALYTIC CONTINUATIONS OF A FUNCTION ELEMENT ALONG ALL CURVES IN A HOMOTOPY EQUIVALENCE CLASS YIELD THE SAME RESULT.

Section 10.4. Riemann Surfaces and Analysis 421

PROOF. From 10.3. 12 it follows that the unique lifts of 1'1 and 1'2 through [II , a] (= [hI , a] ) in W are homotopic and end at the same point. One of the lifts ends at [gJ , b] , whence [gJ , b] = [hK , b] , from which the desired result follows. D

[ 10.4.6 Note. If R is a Riemann surface and F is holomorphic in a neighborhood N (30) �f Q, the analytic continuation of the function element (F, Q) is definable in a manner analogous to that used for defining the analytic continuation of a function element (I, Q) . The corresponding analytic structure AS induces a covering space triple (AS, R, p) .]

10.4.7 Exercise. a) The analog of the Monodromy Theorem holds in the context described in 10.4.6. b) If the Riemann surface R is simply connected, analytic continuation of a function element is independent of the choice of the underlying curve.

10.4.8 THEOREM. IF u IS HARMONIC ON A SIMPLY CONNECTED RIEMANN � F � . SURFACE I�, THERE IS A HARMONIC CONJUGATE v SUCH THAT = u + l V

IS HOLOMORPHIC ON R. PROOF. If 3 E R and {N(3) , q;} is a canonical chart,

- clef - 1 ( ) U = U 0 q; E HaIR 1U ,

and there is a v such that 11 + iii E H (1U) . Thus, if v �f v 0 q;, then

If Q is the set of points \1) t� which f can be continued analytically along a curve starting at 3 , then Q is open and nonempty. Owing to the Monodromy Theorem in the context of R, there is a unique function F holomorphic in Q and such that F IN (3)= f. The analytic continuation may be carried out by means of a sequence of neighborhoods drawn from canonical charts. For each such neighborhood, the argument of the first paragraph shows that �he real part of the corresponding analytic continuation is u: throughou� Q, �(F) = u.

If � E QC, there is some canonical chart {N (�) , 'ljJ}, and again - clef - 1 ( ) U = u 0 'ljJ E HaIR 1U .

- cl f - - cl f -For some V, H � U + iV E H(1U) and if V � V 0 'ljJ, then


In W � Q n N(�) , H - F assumes only imaginary values: 5.3.62 applied to (H - F) 0 'ljJ- l shows that for some constant c, H lw= F lw+c, whence L E Q, i .e., Q is closed (v. also PROOF of 6.2.24b) ) . Since R is connected, Q = R: F is holomorphic on R and 'S(F) is a harmonic conjugate of u.

D

10.5 . The Uniformization Theorem

The current Section covers the concepts and arguments leading to the Uniformization Theorem and its associated statements.

The Uniformization Theorem deals with simply connected Riemann surfaces and their three paradigms, 1U, Coo , and C. Some of the results established in the course of the derivation of the Uniformization Theorem apply more generally to arbitrary-not necessarily simply connectedRiemann surfaces.

In Section 10.6 there are Exercises that lead to the proofs of some of the assertions stated without proof in the outline itself. Since entire books are devoted to the derivation of the Uniformization Theorem, at best, only suggestions of the arguments that constitute its proof are given below. The interested reader is urged to consult [AhS, Be, FK, Fo, Jo, Nev2 , Re, Spr] for further details.

The theme of the development is that for each kind-hyperbolic, elliptic, parabolic-of Riemann surface R there is a nonempty set of functions of some special type - subharmonic, harmonic, holomorphic, meromorphic, or admissible. The last two types are described in

10.5.1 DEFINITION. A FUNCTION F : R r-+ C ON A RIEMANN SURFACE R IS meromorphic IFF FOR EACH CHART {Va , q)a} , F 0 q)� l IS MEROMORPHlC, i.e. , FOR THE CHART z AND SOME k IN Z, F(z) = LCnzn .

n>k A FUNCTION H : R r-+ C IS admissible AT A POINT \1) OF R IFF H IS

BOUNDED OFF EACH NEIGHBORHOOD OF \1) AND H HAS A SIMPLE POLE AT \1) .

10.5.2 Example. When u in Cn i s such that eU (z) = z , for some (local) determination of In, if z E 10, u(z) = In(z) . The function u is meromorphic and admissible at 0, while l u i is harmonic in 10.

The mechanisms and concepts-barriers, Perron families, Dirichlet regions, and Green's functions-of Sections 6.3 and 8.5 are transferred (and modified as needed) so that they are applicable in the context of R. The significant results that lead to the Uniformization Theorem are organized in the following manner.

Section 10.5. The Uniformization Theorem 423

A ) a) For a canonical chart z and r in (0, 1 ) , there is a barrier for the Riemann surface Rr �f R \ { z : I z l ::;; r } at every point of the set Cr � { Z : I z l = r } . b) If I E C (Cr , lR) , in HaJR (Rr) there is a function UJ such that for each a in Cn lim UJ (z) = I(a) : UJ is a solution z-+a

z E 'R r of Dirichlet 's problem for the region Rr and the boundary values I.

�) The Riemann surface R is hyperbolic iff R supports a nonconstant negative harmonic function u iff Green's function G(· , a) exists for some a iff Green's function G(· , a) exists for each a.

C) If R is not hyperbolic (hence is compact or parabolic) , 11.1 E Q c R, and 1 E H (Q \ {11.1} ) , there is a unique function u: a) harmonic On R \ {11.1} ; b ) bounded off each neighborhood of 11.1 ; c ) and such that E)

u - 'iR(f) E HaJR (Q \ {11.1} ) and u(l1.1) = O .

[ 10.5.3 Note. The impact of B) and C) i s the following.

If R is a Riemann surface and K is a nonempty compact subset of R, there is a function 1 harmonic on R \ K.

If K is the (compact) closure of a conformal disc, "( is rectifiable, connecting two points a and b ,

"(* c Ro �f R \ K, and 1 E Ha (Ro) ,

then 1 J dj2 + ( * df) 2 �f dl' (a, b) > 0 and d(a, b) �f inf dl' (a, b) is I' I'

a metric on Ro. This metric is an essential tool in the argument mentioned in 10.3.26 about the proof that every Riemann surface is second countable.

The second count ability of every Riemann surface is a consequence of the Uniformization Theorem, proved by the methods outlined below and which do not establish first that a Riemann surface is second countable, v. 10.6.9.]

D) If a and b are two points on R, there is a function 1 meromorphic on R and such that a E P(f) and b E Z (f) .

The results B)-D) are roughly summarized by: Each Riemann surface R supports a function 1 harmonic at every point save one and that at the exceptional point, say a, 1 has a

1 singularity. Furthermore, if R is hyperbolic, 1(3) - In --- IS 13 - al


harmonic near a, while if R is not hyperbolic,J(3) - � (_1_) is 3 - a harmonic near a.

E) If R is simply connected and hyperbolic, for each chart z and Green's function G, G(z) + In I z l is harmonic and for some F in H (R) ,

G(z) + In I z l = In IF(z) l .

z Furthermore , for each chart z , the function h : R 3 z r-+ F(z) IS a biholomorphic map and h(R) C 1U. (Riemann's Mapping Theorem (8. 1 .1 ) implies that for some conformal map g,

g o h(R) = U) F) If R is simply connected and elliptic or parabolic, and a E R, the

following obtain. Fl) There exist admissible functions at a. F2) If fa resp. fb is admissible at a resp. b, for some Mobius transfor

mation T, fa = T o fb . F3) An admissible function is injective .

• If f is admissible, f(R) C Coo and R is biholomorphically equivalent to Coo resp. C according as R is elliptic resp. parabolic.

The conjunction of E) and F) is the essence of .the Uniformization Theorem. In summary, for any simply connected Riemann surface R:

• R is hyperbolic iff there is a biholomorphic map R r-+ 1U; • R is elliptic iff there is a biholomorphic map R r-+ Coo ; • R is parabolic iff there is a biholomorphic map R r-+ C.


10.6 .1 Exercise. If: a) {(Ii , ni ) } l<i<I is a chain that provides an analytic continuation (iI, nJ) of (iI , nJ ;- b) al E n1 and aI E nI; there . . clef (0) d' clef (1 ) d h h IS a curve "/ startmg at a l = "/ , en mg at aI = "/ , an suc t at I "/

* C U ni . For such a ,,/ there is a chain { [gj , D (aj , rin L::S;j:::::J such i=1

(Analytic continuation can always be achieved by analytic continuation along a curve and via circular function elements.)


10.6.2 Exercise. The Schwarz Reflection Principle (v. 6.2.17 and 8.2. 10) provides a means for analytic continuation. 10.6.3 Exercise. What is a useful and natural definition of a pole at a of a map f : M \ a 3 3 r-+ N between the complex analytic manifolds M and N? 10.6.4 Exercise. The complement C �f 3 \ I (W) (cf. 10.2.47) in 3 of I (W) is discrete, i.e., each point 3 of C is in some neighborhood N(3) containing no other points of C: N(3 ) n C = {3} . 10.6.5 Exercise. If X is curve-connected and h : X r-+ Y is a continuous surjection, h(X) is curve-connected. 10.6.6 Exercise. If X is simply connected and h : X r-+ Y is a homeomorphism, h(X) is simply connected. 10.6.7 Exercise. If

X �f { z �f x + iy -00 < x < 0, -7r < y < 7r } , X �f { z �f x + iy -00 < x < 0, -7r < y ::; 7r } ,

Y �f D(O, It \ (-1 , 0) , Y �f D(O, It \ {O} , h : C 3 z r-+ eZ E C :

a) H � h l x and ii �f h l x are continuous bijections; b) X and X are simply connected; c) H(X) = Y and ii (X) = Y; d) Y is simply connected; e) Y is n�t simply connected: the image under a continuous injection of a simply connected set need not be simply connected. 10.6.8 Exercise. Since 10.6.6 implies H in 10.6.7 is not a homeomorphism, where is ii- I not continuous?

The following items are offered as a guide to the proofs and cOnsequences of the assertions in A)-F) .

For A) : a) The result 6.3.29, proved in 8.6. 16, applies. b ) The methods of

Section 6.3 apply.

10.6.9 COROLLARY. A RIEMANN SURFACE R IS SECOND COUNTABLE. PROOF. In the notations of A ) , f may be chosen to be nonconstant. If (T, RT, p) is the universal covering space triple for RT the function


is harmonic on T. Hence in H (T) there is a nonconstant F such that �(F) = Uj, v. 10.4.8.

Furthermore, if w is a chart for T, then [F(w) , 1U] is a function element for an analytic structure AS of which T is a covering space: T is a covering space of a region Q in C. Since Q is second countable, 10.2 .10 implies T is second countable. Hence Rr and R = Rrl.,J { z : I z l :::; r } are second ��. D

For B): 10.6.10 Exercise. If R i s a Riemann surface, G( · , 11.1) i s a Green's function at 11.1, and k > 0, then 0 > U �f - min{k, G( . , I1.1) } , U is subharmonic, and U is not a constant: R is hyperbolic.

Conversely, if R is hyperbolic, the following items provide the argument for the existence of a Green's function (which the Maximum Principle implies is unique) .

B1) For a in R and a chart z at a, the set

{ w : 0 :::; w E SH (R \ a) , suppw E K(R) , w (z) + In I z l E SH(z) }

is a non empty Perron family. B2) If K �f { z : I z l :::; r } , Q �f R \ K, u E C (QC ) n HaIR (Q) , u l a(n) = 1 ,

I clef ( II 0 :::;

M .

I z l= 1 z _r r - 1 [Hint: The maximum principle implies Vr + In r :::; VI :::; vrMr. ]

B4) If g �f sup v , Perron's method implies g E HaIR(R \ {O} ) . Further:F

{ 1 } { clef 1 In r } . more, 0 0 and if O(z) + In I z l E HaIR [D(O, rt] ' then 0 2': G. Hence the Green's function at a is G.

For C): Sections 5.8 and 5.9 and 6.5.8 provide the basis for the argument.


10.6 .11 Exercise. If R is compact and K �f { z 1 z 1 :::; 1 } is a (closed) conformal disc on R, r < 1 , f E Ha (R \ { z : I z l 2': r } ) and I f(z) 1 :::; M, Stokes's Theorem and Green's formula are applicable:

r * df = r * df = r d * df = o. fa(K) fa(R\K) fR\K

(10.6.12)

The items C1)-C8) below are devoted to establishing (10.6.12) when R, which is not hyperbolic, is also not compact . C1) For the approximants Wn to Uf in the PROOF of 6.3.15, if {p, q} e N,

8p+qW 8PHU then 8 n converge uniformly on compact subsets of Q to f

�x qy �x�y [Hint: 5.3.35 and 6.2.25 apply.]

C2) If R is not compact it contains relatively compact Dirichlet subregions Q such that K is a relatively compact in Q. For each such Q there is a function fn in HaIR (Q n { z : Iz l 2': r } ) and such that

In (z ) = { t(z) if Iz l = r if z E R \ Q '

The set of all such fn constitutes a Perron family F (parametrized by F clef the set of relatively compact regions Q containing K) . If = sup fn

:F and Iz l 2': r , then f(z) = F(z ) . [Hint: The Maximum Principle and the fact that R is neither compact and nor hyperbolic apply.]

C3) If gn E HaIR (Q n { z : Iz l 2': r } ) and { I if l z l = r gn(z) = 0 if z E 8(Q) '

f clef d the set 9 of all gn is, like F, a Perron family. I G = sup gn an 9

Iz l 2': r, then G(z) = 1 . C4) In F resp. 9 there are sequences {fn}nEN resp. {gn}nEN associated

with relatively compact regions Qn and converging together with their derivatives uniformly on compact sets to F resp. G and their derivatives, v. C2) .

C5) 1 gnt:J.fn - fnt:J.gn = 0, n E N, (v. 5.8. 18) . nn\K C6) If 0 < p < r < � and Up E HaIR [R \ D(a, pt] ' while up I Ca (p) = 1R(f)

up is the solution of Dirichlet 's problem for the boundary values 1R(f)then for k (r) as in 6.5.7, max up :::; k (r) . R\D(a,r) O

428

C7)

Chapter 10. Riemann Surfaces

1 . If 5 > rn ..l- 0, for each n there IS a sequence {Unrr,} mEN such that {Unm},nEN ::) {un+ l .m }mEN and for fixed n, Unm converges uniformly on A (0; rn , 1 ) . For every positive r, the diagonal sequence {umm}mEN converges uniformly on A(O; r, 1 ) to a function U harmonic in A(O; r, 1 ) ; the Maximum Principle implies convergence i s uniform on A(O; r , (0 ) : {Umm} mEN converges uniformly on A( 0; 0, (0 ) to a function U harmonic in A(O; O , oo) .

C8) The argument in 6.5.7 leads to I [u - �(f) ] (t�il! ) I :::; k(r) ( 1 � t ) , which converges to 0 as t -+ O.

For D) :

10.6.13 Exercise. For fixed charts z resp. w at a resp. b , i f R i s hyperbolic, Green's functions Ua resp. Ub exist. If R is elliptic or parabolic, there are functions Ua resp. Ub harmonic in R \ {a} resp. R \ {b} and such that ua (z) - � (�) resp. Ub (W ) - � (�) are harmonic near a resp. b .

In either event the function f � (ua)x - i (ua)y

meets the require(Ub)x - i (Ub)y ments of D) .

[Hint: If Ua is a Green's function, for SOme h holomorphic near 0, u (z) + In I z l = �[h(z ) ] . Similarly, if ua (z) - � (� ) is harmonic

near 0, for some g holomorphic near 0, ua (z ) - � (�) = �(g) .

In those respective cases, (ua)x (z) - i (ua)y (z) = h(z) - � resp. 1 (ua)x (z) - i (ua)y (z) = g(z) - z2 : the numerator resp. denomi-

nator of f has a pole at a resp. b .]

For E) :

10.6.14 Exercise. a) For a in R, there is in H (R) an f �f U + iv such that in each chart z, u (z) = G(z) + In I z l , v. 10.4.8. If

F(z) � exp [G(z ) + In I z l + iv(z) ] ,

I z I clef then In I F(z) 1 = G(z ) + In I z l and G(z) = - In F(z) = - In I ha(z) l ·

b) I ha l < 1 .


ha(b) - ha(z) . . I c) For b fixed, ¢ : R 3 Z r-+ ( » IS holomorphlc on R and I¢ < 1 1 - ha b ha(z (v. Section 7.2 and Chapter 8) . d) If Ord f (a) �f

n, off Z(¢) , u �f - '! ln l¢ 1 is harmonic and positive off n

the zeros of ¢. e) If ( is a chart at b, then :F consisting of functions w such that:

el) w is subharmonic and nonnegative off b; e2) supp (w) is compact; e3) w(() + In 1 ( 1 is subharmonic at b, is a Perron family.

f) u (() � G(() �f - In I h b l and hb (a) � 1¢(a) l * � 1¢(a) l . g) 1- == l . hb h) Z(¢) = b and ¢ is injective.

Riemann's Mapping Theorem implies the Uniformization Theorem for hyperbolic Riemann surfaces .

For F) :

10.6.15 Exercise.

Fl)

F2)

F3)

1 For some meromorphic f, g : Z r-+ f(z) - - is holomorphic near a, f Z

is holomorphic off each neighborhood of a, 'iR(f) �f u is bounded, and

u(a) - 'iR (�) = o. F:r some meromorphic 1, g : Z r-+ 1(z) - � is holo

morphic near a, and f is holomorphic and bounded off each neighborhood of a. Near a, both f and 1 are injective. If b is near but different

h clef 1 d - clef 1 h 1 h' ff from a, t en g = ( ) an g = are 0 omorp lC 0 b f - f b f - f(b) while b i s a simple pole of g and ?j. Some linear combination ag + fig is holomorphic and bounded. Since R is not hyperbolic, ag + fig is constant on R: for some Mobius transformation T, f = T (1) . Finally,

1 = if, 2s(f) = -'iR (1) , whence f is admissible. The end of the argument in Fl) implies that the set S of points b such that for some Mobius transformation T, fb = T (fb ) is both open and closed: S = R. If f is admissible at a and f(a) = f(b ) , for a g admissible at a and constructed as in Fl ) , for some Mobius T, g = T(f) , whence g(a) = g(b) . Since a is the unique pole of g, a = b .

F4) If R is elliptic, i.e. , compact, for an f admissible at a, f(R) i s both compact and open, whence f(R) = Coo .


If R is parabolic, and f is admissible, f (R) c Coo and f is not surjective. For some Mobius T, T o f(R) c C. If T o f(R) �C, then T o f(R) is conformally equivalent to 1U: R is hyperbolic, a contradiction.

10.6 .16 Exercise. The fundamental group 7rl (R) of a Riemann surface is finite or countable.

[Hint: The result 10.2.10 and the construction of the universal covering space apply.]

10.6.17 Exercise. a) For some f in H (1U) , if Z E 1U, then

2 exp { [J (z W} = 1 + z ,

i .e. , f is some branch of Jln 1 ; Z and is holomorphic in 1U. b) Analytic continuations of f along the curves 'Y± : [0, 1] 3 t r-+ ±1 + e2rrit lead to different function elements. c) The functionals 'Y± are homologous in Q � C \ ({l} u {-I} ) . d) The Q�homotopy equivalence classes h+} h- } and h- } h+} are different , i.e., h+} h- } h+}-l h_rl -j. 1 . Hence 7rl (Q) is not abelian. 10.6.18 Exercise. Without reference to the Uniformization Theorem, for any Riemann surface, the conjunction 5j 1\ IE is impossible. (Thus the listing of possibilities in the second and third columns of Table 10.3.1 is exhausti ve. ) 10.6 .19 Exercise. If Q c C, f E H (Q) , and for each a in C and some nonempty open neighborhood N(a) , f may be continued analytically to a function element (fa , N( a ) ) , there is an entire function F such that Fln= f.

10.6.20 Exercise. When Q � [0, 1] 2 is regarded as a subset of C, if f E H(Q) and f[8(Q)] C lR, there is an entire function F such that FIQ= f.

[Hint: The Schwarz Reflection Principle applies.]

1 1 Convexity and Complex Analysis

1 1 . 1 . Thorin's Theorem

In a number of disparate contexts, e.g. , topological vector spaces, probability theory (measure theory confined to measure spaces (X, 5, p,) for which X E S and p,(X) = 1 [Kol] ) , von Neumann's theory of almost periodic functions on groups as well as his theory of games [NeuM] , linear and convex programming, etc. , the role played by convexity is central. The following discussion attempts to illustrate that role in complex analysis.

11 .1 .1 DEFINITION. WHEN {Vd 1<k<n IS A SET OF VECTOR SPACES , Fk E [0 , (0) Vk , 1 :::; k :::; n, AND G E ilR � ¢ IN lRV, x " , x Vn IS (G; F1 , " ' , Fn)-

n CONVEX IFF WHENEVER 0 :::; tk , L tk = 1 , AND G o ¢ IS DEFINED,

k=l

n G O ¢ (Vl , . . . , Vn ) :::; L tkFdvk ) '

k=l

[ 11 .1 .2 Note. In the context of Jensen's inequality (3.2.35) , a function ¢ convex in the ordinary sense is (id ; id , id )-convex.

When

V �f U(X ) V �f U'(X ) 1 , p" 2 , p, ,

clef I I I I clef I I I I

clef Fl = In p, F2 = In p' , G = In,

Vi E V;, i = 1 , 2 , f (vl , v2 ) �f I Ix Vl (X)V2 (X) dp,(x) I ' Holder's inequality states that ¢ �f ln f is (G; F1 , F2 )-convex:

431

432 Chapter 11 . Convexity and complex analysis

� 27T kt 1 - e 27Tnni

Since � e-n- = ----,c;2-=-w'- = 0, for h in MVP [D(a, rt] ' n in N, k=O 1 - e �

and s in [O , r ) , � denoting approximately equal to ,

1 2 7r k i [n- l

1 h ( a) = h �:;;: (a + se -n- )

r 0 � 1 ( 2 W k , ) = IIr h (a + st) dT(t) = � :;;: h a + se-n- . ( 1 1 . 1 .3) 1l' k=O

Thus, when = and � are read :::; , ( 1 1 . 1 .3) may be viewed as a convexity property of h. By abuse of language, subharmonic functions, may be viewed as a class of convex functions. ]

When X is a set, B, D c X, and I E jRx , the Maximum Principle in D relative to B obtains for I iff sup I(x) :::; sup I(x) .

xED xEB

11. 1.4 Example. The Maximum Modulus Theorem asserts, i .a. , that if

D(a, r ) C Q and I E H(Q) ,

the Maximum Principle in D � D( a, r t relative to IJ �f 8[D( a, r ) ] obtains for II I . 11. 1.5 Exercise. If D e B, the Maximum Principle in D relative to B obtains for all I in jRx .

11. 1.6 Exercise. If a function I in jRlR is convex or monotone, for every finite interval [a, b] , the Maximum Principle in [a, b] relative to the set {a, b} (consisting of the two points a and b) obtains for I.

11. 1.7 Example. If

I (x) �f { �Os x if ° :::; x :::; 27r otherwise

for every finite interval [a, b] , the Maximum Principle in [a, b] relative to { a, b} obtains for I in jRIR . However, I is neither convex nor monotone: the converse of 11.1 .6 is false.

11 . 1.8 LEMMA. FOR A VECTOR SPACE V, A ¢ IN jRV IS CONVEX IFF FOR EACH A IN jR, EACH PAIR {x, y} IN V, AND EACH t IN [0, 1] , THE MAXIMUM PRINCIPLE IN THE INTERVAL [x, y] �f { z : z = tx + (1 - t)y, t E [0, 1] } RELATIVE TO {x, y} OBTAINS FOR ¢ [tx + (1 - t)y] - At .

Section 11 . 1 . Thorin's Theorem

The map ¢ in lR v is convex iff for each A in lR

sup {¢[tx + (1 - t)y] - At} ::;; max{¢(y) , ¢(x) - A} . O<::t<:: l

PROOF. If ¢ is convex,

¢[tx + (1 - t)y] - At ::;; t¢(x) + (1 - t)¢(y) - At = t [¢(x) - A] + (1 - t)¢(y) ::;; max {¢(x) - A, ¢(y) } .

433

Conversely, if the Maximum Principle as described obtains, for all A in lR, sup ¢[tx + (1 - t)y] - At ::;; max {¢(y) , ¢(x) - A} . Hence

If

tE [O, I ]

sup ¢(tx + (1 - t)y) ::;; max {¢(y) + At , ¢(x) - A + At} O<::t<:: l

clef = max {¢(y) + At , ¢(x) - A(l - t) } = max {A, B} .

max {A B} = { A, then ¢(tx + ( 1 - t)y) ::;; ¢(y) + At , B, then ¢(tx + (1 - t)y) ::;; ¢(x) - A(l - t) .

In each case when A = ¢(x) - ¢(y) ,

¢[tx + (1 - t)y] ::;; t¢(x) + ( 1 - t)¢(y) . D

11. 1.9 Exercise. If each element of {¢,x },XEA is convex, ¢ � sup ¢,X is ,XEA convex. 11 . 1. 10 Exercise. If g in lRlR is a monotonely increasing function continuous on the right and the Maximum Principle in D relative to B obtains for j, the Maximum Principle in D relative to B obtains as well for g o j.

The development above leads to the next result, which implies a number of important conclusions.

11 .1 .11 THEOREM. (Thorin) HYPOTHESIS: a) X IS A VECTOR SPACE; b) C IS A CONVEX SUBSET OF X; c) Y IS A SET; d) j E lRxxY ; e) FOR EVERY LINE SEGMENT

[xo , xd �f { xo + t (Xl - xo ) 0 ::;; t ::;; 1 }

434 Chapter 1 1 . Convexity and complex analysis

CONTAINED IN C AND EVERY A IN JR, THE MAXIMUM PRINCIPLE IN

[xo , xd X Y RELATIVE TO (XO X Y) U (Xl X Y)

OBTAINS FOR FXQ ,Xl ,>' : [0, 1] X Y 3 (t, y) r-+ I [tXl + ( 1 - t)xo , y] - At .

CONCLUSION: M(x) �f sup I (x, y) IS A CONVEX FUNCTION OF X. yEY

PROOF. As formulated, the result is equivalent to the statement that for all A, the Maximum Principle in [0 , 1] relative to {O, I} obtains for

I [tXl + (1 - t)xo , y] - At.

From 11. 1.8 it follows that for each y, I (x, y) is a convex function of x. Then 11 . 1.9 applies. D

11 .1 . 12 THEOREM. IF X AND Z ARE VECTOR SPACES , K C [Z, X] , C IS A CONVEX SUBSET OF X, AND Z E n L- l (C) �f r THEN, IN THE CONTEXT

LEK

ABOVE, M(z; K) �f sup { F(L(z) , y) : L E K, y E Y } IS CONVEX ON r .

PROOF. Since L is linear, L- l (C) is convex. Then 11 .1 . 11 applies. D [ 11 .1 .13 Note. If r = (/), the conclusion above is automatic. In any event, since each L is linear and C is convex, r is convex.

The result asserts in particular that if M(z; K) is finite at both endpoints of a line segment J lying in r, then M (z; K) is bounded above on J.]

1 1 .2 . Applications of Thorin's Theorem

The applications of Thorin's Theorem are not limited to the field of complex analysis. They appear as well in classical functional analysis and in the theory of harmonic and subharmonic functions.

11 .2 .1 Exercise. Thorin's Theorem implies Holder's inequality. In Section 7.3 the focus is on some entire function I and the manner

in which I I(z) 1 grows as I z l t oo. For convenience of description, the growth of various logarithmic functions of I I I rather than the growth of I I I itself is estimated. The THEOREMs that follow are concerned with some function I holomorphic in a region Q that is a strip or an annulus and the behavior in Q of some logarithmic function related to I I I .

Section 1 1 .2 . Applications of Thorin's Theorem 435

11.2 .2 THEOREM. (Hadamard's Three-Lines Theorem) IF

Q �f (a, b) x lR, I E H(Q) n C (Qc , q , I(Q) c D(O, K)

AND M(x; I) � sup { 1 1(x + iy) 1 : -00 < y < oo } , THEN In M IS CON-- d f (-) VEX . PROOF. For each ), in lR, eAz I(z) � F(z) conforms to the hypotheses for I . The Maximum Modulus Theorem implies, in the present context ,

M(x; F) :::; max {eAa M(a; I ) , eAb M(b; I) } and

In M(x; F) = ),x + In M(x; I) :::; max { ),a + In M(a; I ) , ),b + In M(b; I) } . As in the PROOF of Thorin's Theorem, if

In M(b; 1) - ln M(a; I) . . - -), = , I .e . , If ),a + In M( a; I) = )'b + In M(b; I ) , a - b

then direct calculation shows that for t in [0 , 1] ,

In M [ta + ( 1 - t) b; Il :::; t In M ( a; I) + ( 1 - t) In M (b; I) . D

11.2.3 THEOREM. (Hadamard's Three-Circles Theorem) IF

I E H [A(a, r : Rtl n C[A(a, r : R) , ej ,

FOR t I N [r, R] , In M(t; I ) IS A CONVEX FUNCTION OF In t . PROOF . If ° < r < R, a = In r, b = In R, and Q = (a, b ) x lR, for a suitable real 0', the map ¢ : QC 3 z r-+ eC>z carries QC onto A( a, r : R) . Hence I 0 ¢ conforms to the hypotheses of 11 . 1 . 11 . D

[ 11.2 .4 Remark. It is the method of proof of Thorin's Theorem rather than the theorem itself that leads to the last two results . In [Thl the author attributes his attack on his general theorem to Hadamard's original approach!

For In M(x; I) resp. In M(r; I ) , one may substitute g o M(x; I) resp. g o M(r; f) when g is a monotonely increasing function continuous on the right . A notion of the strength of Hadamard's Theorem and of the last remark can be derived from the following considerations.

The Maximum Modulus Theorem implies merely

M(x; I) :::; max{M(a; I ) , M(b; I)}


resp. M(t; I) � max{M(r ; I ) , M(R; I) } .

These inequalities do not imply the convexity of either M(x; I) or M (r ; I) ; a fortiori , they do not imply any of the convexity properties of In M(x; I ) , In M(t; I ) , g o M(x; I ) , or g o M(r ; I ) .

The replacement of, e.g. , M by In M, leads to a function like ¢ in 11 .1 . 1 .]

11 .2 .5 THEOREM. IF I E H (1U) , r E (0, 1 ) , p E (0, 00 ) , AND

a) ON (0, 1 ) , Ip (r) IS A MONOTONELY INCREASING FUNCTION OF r ; b) ln lp(r) IS A CONVEX FUNCTION OF ln r . PROOF . a ) I f ° � r1 < r2 < r , the Maximum Modulus Theorem implies

that for some r2ei02 , � t I I (r1 e 2""wi ) IP � � t I I (r2e 2""W' ei02 ) IP . k=l k=l

b) Thorin's method of proof as exhibited in the PROOF of 11 .2 .2 and as used in the PROOF of 11.2 .3 applies . D

11.2.6 THEOREM. (M. Riesz) IF 0', f3 > 0,

FOR THE (BILINEAR) MAP

Tn n B : em x en 3 (x, y) r-+ L L ajkXjYk �f B(x, y) ,

j=l k=l

THE CONDITIONS

Pj > 0, (J'k > 0, 1 :::; j � m, 1 :::; k � n,

def def ( P = (P1 , · · · , Pm ) , 0" = (J'l , . . · , (J'n ) ,

AND S THE SET OF ALL (x, y, P, 0") SUCH THAT

m n LPj IXj l � :::; 1 , and L(J'k IYk l fr :::; 1 , j=l k=l

Section 1 1 .2 . Applications of Thorin's Theorem 437

THE LOGARiTHM OF Mu(3 �f sup I B (x, y) 1 IS A CONVEX FUNCTION (x.y.p.CT)ES

OF THE (a , /3) IN THE (OPEN ) QUADRANT Q �f { (a , /3) : a , /3 > o } . [ 11.2 .7 Remark. Thus, if

{(I', J) , (1], () ) C Q, 0 � t � 1 , and (a , /3) = t (1', J) + ( 1 - t) (1], () ,

(MO:(3 is a multiplicatively convex function in Q) .]

PROOF. The notations below are useful in the discussion that follows:

For >'1 , ),2 real and fixed and (0'0, /30) in Q, owing to 11 . 1 . 11 and 11 . 1 . 12 , it suffices to prove that the logarithm of

is a convex function of t on (0, 00 ) . If the real variable t is replaced in the right member above by the complex variable t + iu , each ¢j resp. 'l/Jk is translated by U),l In rj resp. U),2 In Sk , and the value of the right member is unchanged:

If all the variabl�save t and u are fixed, Hadamard's Three-Lines Theorem is applicable to M (0'0 + )'It , /30 + ),2t) . Thus In M (0'0 + )'It , (30 + ),2t) is a convex function of t on (0, 00) . D

[ 11.2 .8 Note. a) When 0' = 0 or /3 = 0,

1 1 clef 1 1 clef 00 or - = - = 00. a 0 /3 0


Tn n Concordantly the conditions L IXj I i- :::; 1 or L (J"k I Yk I � :::; 1 are

j=l k=l interpreted IXj I :::; 1 , 1 :::; j :::; m, or I Yk I :::; 1 , 1 :::; k :::; n. The argument given for 11.2 .6 remains valid when a = 0 or f3 = 0, v. 3.2.6.

b) M. Riesz showed that for the bilinear map

BIR : ]R2 x ]R2 3 (x, y) f-t (Xl + X2) Yl + (Xl - X2 ) Y2 , 1 when a = 2 ' the logarithm of the minimum of

is concave. Thorin extended M. Riesz's result by showing that if o < a :::; � , the logarithm of the maximum is concave [Th] . Thus, if B is restricted to ]RTn X ]Rn the conclusion in 11.2 .6 is invalid unless 0' + f3 z 1 and {a, f3} C [0 , 1] ' i.e. , unless (a, f3) E Q, v. Figure 11.2 .1 . ]

11.2.9 Exercise. In terms of

j=l

an equivalent formulation of 11.2.6 is the following.

(0, 1 ) I , 1 )

L-----------------�---- a o

Figure 11 .2 .1 .

Section 11 .2 . Applications of Thorin's Theorem

The map M<>(3 : Q r-+ sup (x,y,p,CT) E S

V (<T .X)7'O

[U(p, X)](3

[V(lT, x)] a

tively convex function of the pair {a, ;3} Q.


439

is a multiplica-

in the quadrant

An important application of 11 .2.6 is found in functional analysis, particularly in the treatment of continuous linear maps

T : £P 3 f r-+ Tf E U, 1 :s; p < 00, 1 :s; q :s; 00.

clef 1 d clef 1 . For such a map, a = - , an ;3 = - , the functlOn p q

should behave like the map M<>(3 considered above. The justification for such an expectation follows.

11 .2 .10 THEOREM. THE FUNCTION Na(3 IS MULTIPLICATIVELY CONVEX IN THE QUADRANT Q.

PROOF. Since the linear span S of the set o f characteristic functions o f sets of finite measure is dense in LP �f LP(Z, p,) , the arguments below about Na(3 when f is confined to S are transferable without change to LP.

If f E S, for some characteristic functions X F ' 1 :s; j :s; m, of pairwise J

disjoint sets Pj of finite measure and some constants X l , . . . , Xrn ,

171 171 rrt f = L XjXFj , Tf = L XjT (XFJ �f L Xjgj .

j=l j=l j=l

There is a set {Ei h:'S: i:'S:n of pairwise disjoint sets of fini te measure such that n

each gj is II I l q-approximable by a gj � L bj;XEi Hence, if ;== 1

'Tn

X; (x) �f L bijxj , 1 :s; i :s; n, j==l


o

11 .2 .11 Exercise. The argument is valid if, for counting measure v and some (Y, S , v) , Lq (� Lq (y, v) .

1 1 clef 1 11 .2.12 Exercise. If 1 :::; q and - + - - 1 = - > 0: a) 1 < q < p' ; b) p q r for some t in (0, 1 ) , r = tp + ( 1 - t) q (v. 4 .9.7) . c) if p = 00 or q = 00, the result remains valid.

The next paragraphs show that (4.9.7) is a direct consequence of 11 .2 .6 and its generalization for No:(3 .

1 1 clef 1 11 .2 .13 LEMMA. (4.9.7 repeated) IF 1 :::; q, - + - - 1 = - > 0, AND P q r

(f, g) E LP(G, p,) x U (G, p,) ,

THEN I l f * g i l T :::; I l f l l p . I l g l l q · PROOF. If {j, g} C Coo (G, q the inequalities

I l f * g l l oo :::; I l f l lp' . I l g l l p , I l f * g l lp :::; I l f l l l . I lg l l p ,

imply that T,q is a map both from LP to Coo (G, q (c L 00) and from LP to LP . The multiplicative convexity of No:(3 and 11 .2 .12 apply. 0

The M. Riesz Convexity Theorem has many consequences . In particular , it leads to the theorems of Hausdorff/Young, and F. Riesz in functional analysis. Since their proofs use the M. Riesz Convexity Theorem for which the proof above involves complex function theory, they appear at this juncture.

The setting is a locally compact abelian group G equipped with Haar measure p,. The central facts relevant to the current discussion are Pontrjagin's Duality Theorem (v. 2) in Section 4.9) and its ramifications in functional analysis. These are treated in detail in [Loo, N ai, We2] .

Section 1 1 .2. Applications of Thorin's Theorem 441

a) According as G is discrete, compact, or neither, i .e . , locally compact and neither discrete nor compact, !.

he dual group 0 of G is compact, discrete, or neither. Furthermore, 0 = G.

b) To each Haar measure p, for G there corresponds a dual Haar measure 11 for O. bI) If G is discrete, L1 (G, p,) C L2 (G, p,) . b2) If G is compact, L2 (G, p,) C L1 (G, p,) . b3) If G is discrete, say G = {g,XLEA' and clef \ A e,X = X{ } ' /\ E , the

g" Fourier transforms tG., ). E A, are a CON for L 2 (0, 11) .

b4) If G is compact, {¢,XLEA is a CON in L2 (G, p,) , and f E L2 (G, p,) ,

then 2 )(1, ¢,X ) 1 2 = l lf(xW dp,(x) (Parseval's equation) . Con-'xEA G

versely, if L i c,X 12

< 00, for some f in L2 (G, p,) , (I, ¢,X ) = c,X . 'xEA

The last is a generalization of the classical Fisher-Riesz Theorem (v. 3.7.14) .

c) If G is neither discrete nor compact , neither of the inclusions

need obtain. However, L1 (G, p,) n L2 (G, p,) �f S is a dense subspace of L2 (G, p,) . If f E S, then 1 E L2 (0, 11) and

(Plancherel's Theorem) . Thus there is definable an extension, again denoted -, to L2 (G, p,) of the Fourier transform and

is an isometric isomorphism (v. 4.9.6 ) . d ) The statement in a ) i s logically equivalent to the conjunction of the

statements in b) and c) : {a) } {} { {b) } 1\ {c) } } . e) If 1 :::; p < 00, by abuse of notation, P � Coo(G, q * Coo (G, q (the

subspace generated in Coo( G, q by functions arising from convolution of functions in Coo(G, C) ) is II l ip-dense in LP(G, p,) . Furthermore, if f E P, then 1 E L 1 (0, 11) and the Fourier inversion formula

442 Chapter 11 . Conve�ity and complex analysis

is valid.

11 .2 .14 Example. The discussion below is based on items a)-e) above and the interpretations and extensions in 5) of Section 4.9. For the map

the previous observations imply

1 8 (I, g) 1 � I l f l l l . I l gl l l , 1 1� 1 2 = I l f 1 1 2 ,

1 8 (I, g) 1 � I I f l 1 2 . I Igl 1 2 .

The multiplicative convexity of No:(3 implies

whence if 1 � p � 2, then 1 1 �lp' � I l f l l p ·

11 .2.15 THEOREM. (Hausdorff/Young and F. Riesz) a ) IF 1 < p � 2 AND THE CON{¢n}nE]\/ (DEFINED ON [0, 1] ) CONSISTS OF UNIFORMLY BOUNDED FUNCTIONS (FOR SOME M, I I ¢n l loo � M, n E N, ) FOR f IN

THERE OBTAIN 1 { cn �f 11 f(x)¢n (X) dX} nE]\/

E £P', AND (� l en IP

') [7 � I l f l l p .

b) IF {cn } nE]\/ E £P , FOR SOME f I N U' ( [0 , 1] , q , 1

11 f(x)¢n (X) dx = Cn, n E N, AND I l f l l p' � (� l en IP) p

PROOF. The discussion of 11 .2.13 suffices for the CON consisting of the functions {e2mrit } nEZ appropriately re-enumerated as a sequence {¢n } nE]\/ ' For the general CON the observations

00

L I cn l 2 � I l f l l � (Bessel's inequality) ,

n=l l en I � Mll f l l l ' (M n-free, )

Section 1 1 .2. Applications of Thorin's Theorem 443

imply that the map T : LP n L 2 3 f r-+ {cn } nEN' regarded as a map from the function space LP n L 2 to the function space eN , is one to which the multiplicative convexity of No:f3 applies. 0

[ 11 .2 .16 Note. As the counterexamples below demonstrate, the condition 1 <z p :::; 2 is essential for the validity of 11 .2 .15.]

1 1 .2 .17 Example. a) If f3 > 1, c E lR \ {O} , and

OCJ icn In n f(x) �f L � e2mrix,

n=2 n'i (In n)f3

then f is continuous but if p > 2, then f I �icn ln n I P' = 00 . n=2 n'i (In n)f3

b) The trigonometric series

is not a Fourier series, i .e. , there is no f in L1 such that

[Zy] . Nevertheless, if q > 2 ,

if m -j. 2n otherwise

00 ( 1 ) q L

n� < 00 .

n=1

( 1 1 .2 . 18)

(After the substitution e27rix -+ Z , the lacunary series in ( 1 1 .2 . 18) is related to the lacunary series in 10.1 .23. )

In a circle of ideas studied by Phragmen and Lindelof, there are generalizations of the Maximum Principle in its various forms for holomorphic, harmonic, and subharmonic functions. Thorin's method implies many of their results. 11 .2 .19 Exercise. A function f in lRfl is subharmonic in the wide sense iff for each subregion Q1 such that Q1 u a (Q1 ) c Q and for each h in

the Maximum Principle in Q1 relative to a (Q1 ) obtains for f - h. [Hint: The argument for 6.3.37 applies.]


11 .2 .20 THEOREM. (Phragmen-Lindelof) IF: a)

Q � { (x, y) : a < x < b, y E lR } ;

b) f E usc (rl) ; c) f E SHW(Q) ; d)

sup f(a, y) � M, sup f(b, y) � M; YE� YE�

e) a < _7r_. f) FOR SOME CONSTANT K , I N rl , f(x, y) � Kealy l ; THEN b - a ' sup f(x, y) � M.

(x,Y)Efl

IF f I S BOUNDED ON 8(Q) AND GROWS AT A CONTROLLED RATE IN Q, THEN f IS BOUNDED IN Q.

7r PROOF. If a < {3 0, and

clef ( a + b) ff (X, y) = f(x, y) - E COS {3 x - -2- cosh {3y, (11 .2 .21 )

the subtrahend in the right member of ( 1 1 .2 .21) is harmonic: ff E SHW(Q) . Then 11 .2 .19 implies that the Maximum Principle in Q relative to 8(Q) obtains for ff ' Since ff � -00 as I y l t oo, the Maximum Principle in Q relative to 8(Q) obtains as well for f· 0

[ 11 .2 .22 Note. If

then

f(x, y) = cos -- x - -- cosh --y, clef 7r ( a + b) 7r

b - a 2 b - a

7r w l Y I f(x, y) � cosh -b -y < e b - a , - a

f(a, y) = f(b, y) = 0,

{a < x < b} ::::} { lim f (x , y) = oo} . ly l-+oo

Hence, if the condition e) in the statement of 11 .2.20 is relaxed 7r to a � -- , the conclusion is false.] b - a

1 2 Several Complex Variables

1 2 . 1 . Survey

The discussion of complex analytic manifolds, v.Section 10.2, provides an introduction to the possibilities of studying analytic functions of several complex variables. The particular case, when n > 1 and Q is a subregion of en , of functions in e� and analytic in Q is of great interest. At first blush the theory seems to be a simple generalization of what is known about holomorphy as introduced in Chapter 5. A closer look reveals that parts of the more extended theory are intrinsically different from their cCcounterparts; furthermore, some of the most useful theorems about elC have no natural extensions to elC

n•

The typical element z of en is a vector (ZI , . . . , zn ) and when en is viewed as ]R2n , then Zj �f X2j- l + iX2j , 1 :::; j :::; n. When Q c en , a function f is in C1 (Q) iff each partial derivative B

B f , 1 :::; k :::; 2n, exists Xk and is continuous.

The concept of analyticity can be introduced via the introduction of the following operators (cf. 5.3.5) :

12.1.1 DEFINITION. A FUNCTION f IN C1 (Q) IS HOLOMORPHIC IN Q, i .e., f E H (Q) , IFF 8f = O.

[ 12 .1 .2 Remark. The condition 8f = 0 is the n-variable version of the Cauchy-Riemann equations.]

When n > 1, the study of functions holomorphic in a region of en IS simplified by the introduction of specialized vocabulary and notation.

445

446 Chapter 12. Several Complex Variables

When Ql , . . . , Qn are regions in c., their Cartesian product

is a polyregion . When each Qk is an open disc D (ak , rkt , Q is a polydisc. clef ) ( + )n �f ( ) �f When a = (a1 , • • • , an E Z , z - Zl , . . . , Zn , a - (a l , . . . , an ) ,

clef ) and r = (rl , . . . , r n ,

a clef "'1 '" clef Z = Zl . . . Zn n , aa = a"" , . . . ,"'n '

clef I I

clef a! = al ! ' " an ! , a = 0'1 + . . . + an ,

n a clef II '" r == ri t ,

i= l

a clef a"" a"'n clef ala i a J =

a "' 1 . . . a "'n = a "' 1 a an ' Zl ' Zn Zl ' " Zn

12.1 .3 Exercise. For a polydisc b.(a, rt, how do

aD [b.(a, rtl and a [b.(a, rtl

differ? Many of the theorems about functions in CC have, when n > 1 , their

natural counterparts for functions in c.cn • There follows a systematic listing of these counterparts.

12 .1 .4 THEOREM . IF J E C[b.(a, r) , C] AND, AS A 'FUNCTION OF zk , WHILE THE OTHER COMPONENTS OF (Zl , . . . , Zn ) ARE HELD FIXED,

THEN IN b.(a, rt,

1 1 J(w) J(z) = -- dWl · · · dw . (27ri )n ao[�(a,r) O l (Zl - wI ) · · · (Zn - wn ) n

12.1.6 COROLLARY. IF J E H (Q) , THEN J E C=(Q, q .

( 12 . 1 .5)

Section 12. 1 . Survey 447

12.1 . 7 THEOREM . IF I E H (Q) , K (Cn ) 3 K c Q, AND 0 (Cn ) 3 U ::) K, THERE ARE CONSTANTS Co. SUCH THAT

12.1 .8 COROLLARY. IF {In}nE]\/ C H (Q) AND FOR EACH COMPACT SUBSET K OF Q, In I K� 1 1 K ' THEN I E H (Q) .

12.1 .9 COROLLARY. THE n-VARIABLE VERSION OF VITALI'S THEOREM (5.3.60) IS VALID.

12.1 .10 THEOREM . IF I E H [b.(a, rt] ' THEN

THE SERIES CONVERGES UNIFORMLY TO I IN EVERY COMPACT SUBSET OF b.(a, rt .

12 .1 .11 THEOREM. (Cauchy's estimates) IF I E H (b.(0, rt) AND

I I (z ) 1 � M IN b.(a, rt,

12 .1 .12 THEOREM. (Schwarz's lemma) IF I IS HOLOMORPHIC IN A NEIGH�ORHOOD OF b.(O, r) AND I I (z) 1 � M IN b.(O, r) , FOR SOME k IN Z+ AND ALL z IN b.(O, r) , I I(z) 1 � M I � l k .

12.1 .13 THEOREM. (Jensen's inequality) IF b.(0, r) C Q, I E H (Q) , AND 1(0) -1 0, THEN A ( \ ) )

1 In I I (z) 1 dA2n � In 1 1(0) 1 · 2n b. 0, r �(O,r)

[ 12.1 .14 Remark. When n > 1, Jensen's inequality implies that if I E H (Q) and I =t=- 0, then

A2n ({ z : z E Q, I(z) = O } ) = O .

Results like (* ) above are valid for holomorphic maps between complex analytic manifolds, v. [Gel] . ]

When n > 1 , there are theorems that have no nontrivial counterparts when n = 1 .


12.1 .15 THEOREM. (Hartogs) IF n > 1 , Q c en , I E en , AND AS A FUNCTION OF EACH VARIABLE Zk , AS THE OTHER COMPONENTS OF (Z1 , . . · , Zn ) ARE HELD FIXED , I IS HOLOMORPHlC, THEN I E H (Q) .

[ 12 .1 .16 Remark. The contrast between 12 .1 .15 and 12 .1 .4 deserves attention.]

n PROOF. (Sketch) a) The formula in (12 .1 .5) implies that if II ri > ° and I I I

i= 1 n is bounded in a polydisc Q �f II D (0, rit , then I is continuous, whence

12 .1 .4 applies. n

b) An argument relying on Baire categories shows that if II ri > ° and i= 1

D(a, r) c Q, then D(a, r) contains a polydisc l1 such that l10 -j. (/) and I I I is bounded in l1.

c) Mathematical induction, Cauchy's estimates (12.1 .11 ) (for the case of n - 1 variables) , and 6.3.39 conclude the proof [Ho] . 0

n 12.1 .17 THEOREM. (Polynomial Runge) IF Q �f II Qk IS A S IMPLY CON

k=l NECTED POLYREGION IN en AND K IS A COMPACT SUBSET OF Q, EACH I IN H (K) IS UNIFORMLY APPROXIMABLE ON K BY POLYNOMIALS .

[ 12 .1 .18 Note. If b is sufficiently small and positive and Q is the polyregion [D(O, 1 + bt] 2 X D(O, bt 3 (Z1 ' Z2 , Z3 ) , the map

is injective and holomorphic. Nevertheless, on F(Q) Polynomial Runge fails to hold [Wer] .]

On the other hand, there are no generally valid counterparts to removable singularities (cf. 5.4.3) nor to the phenomenon of natural boundaries as exemplified by 7.1.28 and 7.1 .29.

12.1 .19 THEOREM. IF n > 1, Q IS A REGION CONTAINING THE BOUNDARY OF D(a, rt, AND I E H(Q) , THERE IS IN H (D(a, rt) A UNIQUE 1 SUCH THAT llnnD(a.r)O = I lnnD(a.r) o ' PROOF . (Sketch) For z in D (a, rt the formula

�( ) 1 1 I (Z1 , . . . , Zn- 1 , W )

I z = - dw 27ri Iw-an l=r W - Zn

Section 12. 1 . Survey 449

defines the f as described. o

12.1 .20 COROLLARY. IF n > 1 , r > 0, AND a IS AN ISOLATED SINGULARITY OF AN f HOLOMORPHIC IN D(a, rt \ {a} , THEN a IS A REMOVABLE SINGULARITY OF f.

ALL ISOLATED SINGULARITIES ARE REMOVABLE, CF . 5.4.3 .

12.1 .21 THEOREM. IF 1 � k < n, E > 0,

S �f { Z '. I I I I I I I I } Zl = . . . = Zk = 1 , Zk+ 1 < 1 + E, . . " Zn < 1 + E , Q ::) D(O, 1t u S,

clef ( ) r1 = . . · = rk = 1 , rk+ 1 = . . · = rn = 1 + E, r = r1 , . . · , rn ,

AND f E H (Q) , THERE IS IN H [Q U D(O, rn A UNIQUE f SUCH THAT lln= f· PROOF. (Sketch) The formula

defines a function as described. o There arises the question of characterizing an open set in en as a

domain of holomorphy , i .e . , roughly described, an open set U for which some f in H (U) has no holomorphic extension to a proper superregion U1 , cf. 7.1 .29.

12 .1 .22 DEFINITION. AN OPEN SET U IN en IS A domain of holomorphy IFF FOR no OPEN SETS U1 , U2 :

a) (/) -j. U1 C (U2 n U) ; b) U2 IS CONNECTED AND U2 ct U; c) WHENEVER f E H (U) THERE IS IN H (U2 ) A (NECESSARILY UNIQUE)

h SUCH THAT f l u! = h lu! '

12.1 .23 DEFINITION. FOR A COMPACT SUBSET K OF AN OPEN SET U, THE H (U) -hull OF K IS

Ku �f { z : z E Q, U E H (U)} '* { I f(z ) 1 � s�p I f (z) l } } .

[ 12 .1 .24 Note. The set Ku is closed but need not be compact.]


12.1.25 THEOREM. IF U IS OPEN IN en , THE FOLLOWING ARE EQUIVALENT: a) U IS A DOMAIN OF HOLOMORPHY; b) IF K IS A COMPACT SUBSET OF U, THEN Ku IS RELATIVELY COMPACT

IN U; c) SOME f IN H (U) HAS NO ANALYTIC CONTINUATION BEYOND U, i .e. ,

THERE ARE NO U1 , U2 CONFORMING TO a) AND b) IN 12 .1 .22 .

12.1 .26 COROLLARY. IF U IS CONVEX , U IS A DOMAIN OF HOLOMORPHY.

12.1 .27 DEFINITION. THE SET OF ALL POLYNOMIALS IS Pol. A Runge domain U IS A DOMAIN OF HOLOMORPHY SUCH THAT IF f E H (U) , THEN f IS UNIFORMLY APPROXIMABLE ON EACH COMPACT SUBSET OF U BY ELEMENTS OF Pol. WHEN K IS COMPACT,

WHEN K = K, K IS polynomially convex.

12 .1 .28 THEOREM. A DOMAIN OF HOLOMORPHY U IS A RUNGE DOMAIN IFF FOR EACH COMPACT SUBSET K OF U, K = Ku .

Further details can be found in [Ho] and [GuR] . They provide extensive treatments of the results cited above and relate them to the theory of partial differential equations, the study of Banach algebras, complex analytic manifolds, etc.

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Symbol List

The notation a.b. d indicates Chapter a, Section b, and page d.

a.e. : 2.2. 60 A \ B: for two subsets of a set X, { x x E A, x tJ. B }, 1 . 1 . 3 A(a; r, R) : 5.4. 234 A(a : r, R; 8, 4;) : 5.4. 234 [B, F] c : 3.3. 105 Bn(f) : 3.2. 100 BP(x, r): 1 .2 . 10 BW: 3.4. 112 (Bft

': 3.4. 112

c : 1 . 1 . 5 Sz : 10.2. 395 Cabed : 8.4. 359 c : 2.2 . 69 C: 1 . 1 . 4 cn : 1 . 1 . 5 C= : 5.6 . 252 C* : 4.9. 192 C: 8.4. 359 CF: 3.5. 129 Ck(C, lR) : the set of functions f having k continuous derivatives , 5 . 1 . 203 C=(C, C ) : the set of functions f having k continuous derivatives , k E N,

5 .1 . 203 Co : 1 .2 . 13 Co(X, C) : 3. 1 . 90 Cu (r): 8.2. 344 Coo(X, C) ) : 2.3. 61 Coo (X, lR) : 2 .3 . 76 Cubed : 8.4. 359 Ck,Tn : 4.6. 165 CON: 3.2. 97 Cont (f) : 3.7. 134 Conv (A) : of a set A, the intersection of the set of all convex sets containing

A, 3.7. 136 cos: 2.4. 80 div : 6.3. 287 DLS: 2 . 1 . 50 D: 2.2. 59 D( a, r ) o : 5 .1 . 203

455

456

D: D (L 1 ) , 2.2. 56 D(L): 2.2 . 56 D(f) : 9 .1 . 369 D(O, Rt : 5.5 . 234; 10: 5.4. 251 det : determinant, 4.7. 183 dP,' dP: 5.8. 261 diam (S) : 1 .2 . 15 dim : dimension, 3.4. 112 Discont (<p) : the set of discontinuities of <p, 1.2. 13 D+ , D- : 3.2. 102 Dl , Dl , Dr, Dr : 4.6. 173 DN : 3.7. 133 ei : ith element of the standard basis for en , 1 .4. 31 exp: 2.4. 80 E: 8.2. 343 E: 5.3. 218 E(f) : 5.4. 231 E - F: 2.3. 78 Et:J.F: 2.5. 87 EN (z) : 7.2. 317 Eo : 2 .1 . 59 fip: 1 .2. 12 f[a] (x) : 4.9. 188 fx : 4.4. 153 J'(a) : 4.7. 178 f(x+) : 4. 10. 196 f(x-) : 4.10. 196 f * g: 4.4. 157 f* : 4.10. 196 f. : 4.10. 196 f -< V: 1 .2 . 15 [j, a] : 10.2. 391 [j, V] : 10.2. 393 (f : g): 4.9. 187 F: 1.2. 7, 1.5. 32, 4.2. 146, 5.9 . 267, 6 .3 . 290, 8 . 1 . 337, 9.3. 374 FN : 3.7. 133 FTA: 5.3. 218, 9 . 1 . 369 FTC: 4.6. 164, 5.8. 262 grad : 6.3. 287

Symbol List

g o f: for f in yX and g in ZY , the map X 3 x r-+ f [g (x)] E Z: 1 .2 . 7 Sp(g, h) , sp (g, h) : 4.6. 170 Gli : the intersection of countably many open sets , 1 .7. 40 GW" W2 : 8.4. 357 id : the identity map: 1 .4. 21 , 3.5. 117 H1 (Q, Z) : 5.5 . 248

Symbol List

Ha(Q) : 5.3. 227 HaJR (Q) : Ha (Q) n IRQ , 5.3. 227 H (A) : 2.2. 68 Hb (Q) : 8.1 . 337 H (S) : 5.3. 209 'S(z): 1 . 1 . 4 IT: 1 .3. 18 I : 10.2. 398 im (T) : 3.6. 129, 4.8. 185 ind -y(a) : 2.5 . 83 Ind -y (c) : 5.2 . 206 J: the DLS integral, 2 . 1 . 50 J(I) : the Jacobian matrix of f, 4.7. 182 K -< f: 1 .2. 15 K(X) : 1 .2 . 12 KG,\: 2.3. 77 Ku: 12. 1 . 449 ker (T) : 3.6. 129 £k : 2.1. 55 £(-y) : length of the curve ,,( , 4.8. 185 lim , lim 1.2 9, 2 .5 . 86 Iz : 2.4. 81 L: the function lattice at the heart of DLS theory, 2 . 1 . 43 LCTVS: 3. 1 . 89 Ld : £1 , 2 .1 . 55 Lf: 4.6. 167 LRN: 4.2. 144 lsc(S) : 1 .2 . 9 Lu: 2 . 1 . 48 Lui : 2 . 1 . 52 £ 1 : 2 .2 . 62 £: 2.4. 81 mid (a, b, c) : 2.2 . 55 M: 8.2. 343 M1 : 8.2. 343 M(a, r) : 5.3. 217 M(R; I) ; 7.3. 323 Mod: 8.4. 358 Mo: 8.2. 343 M(Q) : 5.4. 231 MON: 4.6. 170 NBV: 4. 10. 197 N: 2.2. 62 n � n': 1 .2. 1 1

457

458

0: orthogonal, 3.2. 96 0: 1 . 1 . 5, 3 . 1 . 89 ON: 3.2. 96 Ord f(a) : 5.4. 241

p' : for p in [1 , 00 ), p' �f { � � 1 Pol: 12. 1 . 450 P(k) : 6.2 . 275 P(f): 5.4. 231 Pr((}) : 6.2. 274

if 1 < p 2 2 , 3 . . 9 if p = 1

Symbol List

s.p(X) : the power set of the set X, i.e. , the set of all subsets of X: 1 .2 . 6 p . A: 10.3. 414 Q: 1 . 1 . 3 Qpqrn : 7. 1 . 302 Qk,rn : 4.6. 165

00 Rc , Rs : for the power series S �f L Cn (z - a) n and c �f {cn } nE]\/+ ' the

n=O radius of convergence r" , q.v. : 5 .2 . 205

Rr: 8.4. 359 JR.: 1 .2 . 3 R([O, 1] ) : 2 .2 . 75 �(z): 1 . 1 . 4 JR." : 1 . 1 . 5 ra : 10.1 . 381 S(k) : 5.2 . 206 (X, S , /-l) �f (Xl X X2, SI X S2 , /-l1 x /-l2 ) : 4.4. 153-154 S X K : 4.4. 153-154 S0K : 4.4. 153-154 S: 10.2. 393 sn+l : 5 .6 . 253 sgn : 2.4. 81 SH : 6.3. 285 SHW: 6.3. 293 sp(x) : 3.5. 120 Sp (A) : 3.5. 124 Sa,o: : 10.3. 411 S8 : 8.3 . 351 Sch : 8.2. 347 S(U): 10.2. 393 supp : 1 .2 . 15 Tabed : 8.2. 343 T: 1 .2 . 6 T A : 1 .2. 6 T': 3.4. 116

Symbol List

'][': 1 . 1 . 5 Tf(x) : 4.10. 197 TVS: 3.1. 89 Ua, R : 6.3. 286 usc( S) : 1 .2. 9 1U: 5.3. 224 V': 3.3. 104 V* : 3.3. 104 (Va , 4>a ) : 10.3. 416 [V] C : 3.3. 104 X-I : 3.5. 117 (Xo , . . . , Xn ) : 1 .4. 21 [Xo , . . . , Xn] : 1 .4. 21 (X, d) : 1.2. 8 (X, Y, p) : 10.3. 409 yX : the set of all maps f : X H Y, 1 .2 . 7 { z : I z l < r } : 10.3. 416 3: 1.2. 9, 10.2. 396 ,, : 10.2 . 396 ,6(X) : 3.7. 136 "(

* : 2.4. 79 "( "'A J: 1 .4. 20 "( '" J: 1.4. 20 "( "'F,A J: 1 .4. 20 ;y, r: 5.5. 237 l1u: 5.3. 226 Ja : 2 . 1 . 55 J(a, b) : 5.6. 252 J(F) : 1 .2 . 14 (: 5.9. 269 8 : 5.6 . 251 I-l([a, b] ) : 4.10. 196 I-l X K : 4.4. 154 1-l0K : 4.4. 154 I-l

*, I-l* : 2.2. 68

v: 3.2. 97 n+ : 5.7. 254 7r(x) : 5.9. 269 7rl (Y, Yo) , 7rl (Y) : 5.5. 245 p(T) : 4.7. 178 P(x) : 3.5. 122 L:2 : the Riemann sphere, 5.6.251 a (B, B') , a (B', B) : 3.4. 1 12 a�k) : 1 .4 . 27

459

460 Symbol List

T: 3.5. 127 Q: the equivalence class containing the well-ordered set of all ordinal num

bers corresponding to countable sets, 1 . 1 . 5; a region contained in C, 5 .1 . 204

w(k) , *w (k) : 5.8. 258-259 No : 1 . 1 . 5 (0, Q) : 2.2. 67 #: 1 . 1 . 5 (aj; (X) ) : 4.7. 182 aXj l�i .j<::n 0, QSJ: 4.4. 154 XE: 2 . 1 . 54 �: isomorphic, 5.5. 245 =: equal almost everywhere, 2 .2 . 62 8, 0: 4.4. 151 EB : direct sum, 3.6. 129 8: 5.3. 186 a: 1 .2. 11 , 5 .3 . 210 aoo(Q): 5 .6 . 252 --< : 1.2. 10 A x B: for sets A and B, the set { (a, b) a E A, b E B } , 1 . 1 . 3 X: 4.4. 153 �: converges uniformly, 1 .2 . 17 m-=rs: converges in measure, 3.7. 134 I I�p : converges in the I I l i p-induced topology, 3 .1 . 91 varh(P) 4.6. 169 j l\ g, j V g: 1 .7. 39 {A 1\ B}: logical and, i.e., for the assertions A and B, the assertion A and

B: 2 .1 . 48 ALJB, U A), : unions of pairwise disjoint sets {A, B} , {A)'hEA : 2.2 . 61 ),EA (P, Q): 3.2 . 100 ( , ) : 3.2. 96 I I l i p , I I 1 100 : 3.2. 92-94 I I I I Q : 3.5. 106 a(u, v) a(x, y)

: 5.8. 260 �n : homologous, 5 .5 . 249 �: approximately equal to: 1 1 . 1 . 432 -,G: the denial (negation) of G: 10.3. 417 pl q: for p and q in ;2;, their greatest common divisor: 1 .7. 39 0: the empty set, i .e. , for any set X, 0 = X \ X, 1 .2. 6 l x J resp. r xl : for x in JR., the greatest integer not greater than x resp. the

least integer not less than x: 7.3. 325

Glossary jlndex

The notation a.b. d indicates Chapter a, Section b, page d.

ABEL , N. H . : 4 .10. 193 Abel summation: 4.10. 193

A

abelian: of a group G that the binary operation is commutative: ab = ba, 4.5. 160

abelianization: For a group G, the subgroup generated by the set of all elements of the form xyx- 1 y- l is a normal subgroup, the commutator subgroup C. It is the smallest normal subgroup modulo which G is abelian. The quotient group G /C is the abelianization of G. 5.5 251

absolute topological property: a topological property P such that if (X, T) is a topological space, S e A e X, then S has property P with respect to T A iff S has property P with respect to T, 1 .2 . 17

absolute value: 1 . 1 . 4 absolutely continuous component: 4.2. 144 --- --- function: 3.2. 96, 4.6. 148, 4.6. 171 --- --- measure: 4.2. 143 --- summable: 3.3. 1 1 1 absorbent: 3.4. 1 13 action-invariant : for a set X and a group G acting on X, of an attribute A

of subsets of X that if A obtains for E and g E G, then A obtains for g(E) , 6.2. 274

addition-distributive: for a vector space V, of a map · : V2 '3 (x, y) H X . Y that X · (ay + bz) = ax · y + bx · z, 3.5. 1 16

adjoint: 3 .5 . 1 16 admissible (function on a Riemann surface) : 10.5. 422 advertible: 3.5. 118 AHLFORS , 1 . V . : 9 .1 . 370 ALAOGLU , L: 3.4. 100 Alaoglu's Theorem: 3.4. 115 ALEXANDER, J . W. : 7. 1 . 305 algebra: a vector space A (over a field) on which there is defined an addition

distributive map . , v.addition-distributive, 6 .5 . 296 algebra-homomorphism: between algebras, a map that commutes with the

algebraic operations : 3.5. 124 almost everywhere: 2.2. 62

461

462 Glossary /Index

almost periodic function (on a group G): a map f : G H e such that for each g in G, the closure of U f(ag) is compact , 1 1 . 1 . 431

aEG alternating process: a technique for deriving from the solutions of Dirichlet 's

problem for each of two intersecting regions a solution of Dirichlet 's problem for the union of the regions [Nev2] : 10.4. 418

analytic: 10.2. 404 --- continuation: 5.4. 232, 10. 1 . 382

along a curve "/: 10.4. 419 --- structure: 10.2. 395 analytically equivalent : 10.2. 403 anharmonic ratio: 8.2. 342 annular sector, annulus: 5.4. 234 antipodal point theory: for Sn �f { x : x E ffi.n+1 , I I x l 1 2 = I } , the study

of maps T : Sn H ffi. such that for some x, T(x) = T( -x) , 1 .4. 20 approximate identity: 6 .5 . 296 arc: 2.4. 82 Archimedean: 1 . 1 . 4 arc-length: for a metric space (X,'d) and a curve "/ : [0, 1] '3 t H ,,/(t) E X ,

6 .2 . 274 Argument Principle: 5.4. 243 ARZELA, C . : 1 .6 . 32, 9 .1 . 370 Arzela-Ascoli Theorem: 1 .6 . 37, 9 . 1 . 370 ASCOLl, G . : 1 .6 . 37, 9 .1 . 370 associate (of a Tabed : 8.4. 359 atlas: 10.2. 403 auteomorphism: for a topological space X, a bicontinuous bijection

X H X, 3.5. 119 autojective: of a map X H X that it is bijective, 1 .2 . 7 Axiom of Choice: If { Sa : a E A } is a set of sets, some set S has exactly

one element in common with each Sa , v.Zorn's Lemma, 1 .5 . 32

BAIRE, R . : 1 .3. 19 Baire category: 1 .3. 19 --- set: 2.3. 76 --- space: 2 .3. 75

B

---'s Category Theorem: 1.3. 19 BANACH , S . : 3 . 1 . 89 , 3 .3 . 1 10, 3 .5 . 116 , 3 .5 . 124

Glossary /Index

Banach algebra: 3.5. 1 16 --- field: a Banach algebra F that is a field in which the map

is continuous, 3.5. 124 �-� space: 3 .1 . 89

--- -Steinhaus: 3.3. 110 barrier at a: 6 .3 . 291 barycenter, barycentric subdivision: 1.4. 23 base: 1 .2 . 6 --- of neighborhoods at a point : 1 .2. 8

��-- for a space: 1 .2 . 8 BERGMAN, S . : 8.3. 353 Bergman's kernel: 8.3. 353 BERNSTEIN, S . : 3.2 . 100 Bernstein polynomials: 3.2 . 100 BESSEL, F. W . : 3.2. 97 Bessel's inequality: 3.2. 97

463

bicontinuous: of a map T : X H Y between topological spaces, that T is injective and both T and T- 1 are continuous, 1 .7. 39

biholomorphic: of a map H : X H Y between complex analytic manifolds, that it is injective and that both H and H-1 are holomorphic, 8. 1 . 341 , 10 .2 . 403

bijection, bijective: 1 .2 . 7 binary markers: 1 .2 . 16 biorthogonal pair: 3.3. 108 Birkhoff, G. D . : 4.6. 173 BLASCHKE, W. : 7. 1 . 301 , 7.2. 320 Blaschke product : 7. 1 . 301 , 7.2. 320 BLOCH, A . : 9 .1 . 370, 9.2. 371 Bloch's Theorem: 9.2. 371 BONK, M . : 9 . 1 . 370 BOREL , E . : 2.2 . 59 Borel measurable: 2.2. 59 --- subset : 2 .2 . 59 boundary: 1 .2. 1 1

value problem: 6.3. 288 bounded (set in a topological vector space) : 3 .7. 136 --- variation: 4.6. 170 bounding cycle: 5.5 . 249 branch: 10.2. 391

point: 10.2. 401 --- ---- of order k: 10.2. 399 BROUWER, 1. E. J . : 1 .4 . 25, 1 .5 . 31, 2 .5 . 83

464

Brouwer degree of a map: 1 .4. 25, 2.5 . 83 Brouwer's Fixed Point Theorem: 1 .5 . 31

c

C-ALGEBRA: an algebra over C, 3.5. 1 16 C-vector space: a vector space over C, 3. 1 . 89

Glossary /Index

calculus of variations: the study of extrema (maxima, mlmma, saddlepoints) of functionals, e.g. , for a vector space V, maps V H JR., 8 .1 . 341

canonical atlas, canonical chart : 10.2. 405 --- product : 7.2. 318, 7.3. 328 --- --- representation: 7.2. 318 CANTOR, G . : 1 .2. 13, 2.2. 66-67 Cantor function: 2 .2 . 67 --- set: 2.2. 66 CARATHEODORY, C . : 2.2 . 69 Caratheodory measurable: 2 .2 . 69 cardinality: for a set X, the set of all sets Y such that for some bijection

bxy , bxy (Y) = X, 1 . 1 . 5 Cartesian product : 1 . 1 . 5 CASORATl, F . : 5.4. 230 category: a complex consisting of: a) a class C of objects , A, B, . . . ; b) the

class of pairwise disjoint sets [A, B] (one for each pair {A, B} in C x C) of morphisms; c) an associative law

0 : [A, B] x [B, G] ", (f, g) H g o f E [A, G] �f [A, B] 0 [B, G] ;

of composition of morphisms; d ) for each A in C, in [A, A] a morphism l A such that if f E [A, B] and g E [G, A] , then f 0 l A = f and l A 0 g = g, 3.5. 125, 4.9. 189

CAUCHY, A . 1. DE: 1 .2 . 8 , 1 .6 . 35, 5.3. 21 1-212 , 5.3. 215 , 12. 1 . 447 Cauchy estimate: 5.3. 218, 12 .1 . 447 --- -Hadamard Theorem: 5.2 . 205 --- net: 1 .6 . 35 --- -Riemann equations: 5.3. 210 --- 's formulre: 5.3.211 --- 's integral formula, basic version: 5.3. 215 --- ---- ----, global version: 5.4. 235

--- sequence: 1 .2 . 8 --- --- Theorem: 5.3. 212 ---- ---- Theorem, basic version: 5 .3. 212 tECH , E . : 3 .7 . 136

compactification: 3.7. 136 central index: 7.3. 324

Glossary /Index

chain: 10. 1 . 382 character group: 4.9. 189 characteristic function: 2.2. 59 chart: 10.2. 402 circled: 3.4. 112 circular: 10 .6 . 424 closed ball: 1 .2 . 8

curve: 4.8. 185, 5.2. 182 ---- degenerate n-simplex: 1 .4 . 21 ---- interval: 1 . 1 . 4 --- set: 1 .2 . 1 1 --- simplex: 1 .4. 21 closure: 1 .2 . 1 1

operation: 1 .7. 42 coarser: 1 .5 . 32 cofinal diset: 1 .2. 1 1 --- net: 1 .2 . 1 1

465

commutator: in a group, the product of elements each of the form xyx-1 y- l : 5.5. 250

---- subgroup: 5.6. 251, v.abelianization compact: 1 .2 . 12 complete analytic function: 10 .2 . 391 ---- atlas : 10.2. 403 ---- measure space: 2 .2 . 62

metric space: 1 .2. 8 --- ordered field: 1 . 1 . 3 ---- orthonormal set: 3.2. 97 --- a-ring: 2.2. 62 completion of a measure space: 2.2. 62

of a metric space (X, d) : for the set CS of Cauchy sequences {xn } nE]\/ in X and the equivalence relation ", defined by

the set of ",-equivalence classes; if � resp. 1] is an equivalence class containing {xn } nE]\/ resp. {Yn } nE]\/ the distance between them is

D(�, 1]) �f inf d (xn , Yn ) , 2.5. 87 nE]\/ complex analytic manifold: 10.2. 405

conjugate: 5 . 1 . 204 --- curve: 10.2. 395 --- integration: 5 . 1 . 204 --- measure: 4 . 1 . 137

466 Glossary /Index

--- --- space: 4 .1 . 138 --- numbers: 1 . 1 . 4 --- projective line: 5.7. 254 --- rational point : 10.2. 395 component : in a topological space X, a maximal connected subset of X,

2.5. 83, 5.2. 208 concave: of a map f : V H JR. of a vector space V into JR., that -f is convex,

q.v. , 11 .2 . 438 Conformal Mapping Theorem: 8 . 1 . 336 conformal disc: 10.3. 416 --- self-map: 8 . 1 . 341 conjugate bilinear: for a vector space V over C, of a map

f : V2 '3 {x, y} H f(x, y)C,

that f(x, y) = f(y, x) and f(ax + (3y, z) = af(x, z) + (3f(x, z ) , 3.6. 128

---- -linear: of a map T bet.ween vector spaces , that

T(ax + by) = aT(x) + bT(y) , 3.6. 128, 5.8. 261

--- pair: 3.2. 92 connected: 1 . 1 . 3, 1 .2 . 1 1 continuous: 1 . 2 . 7 --- on the right : of a function f in JR.IR , that for all x, lim f(y) = f(x) , ytx

11 . 1 . 433 contour: 5.7. 254 converge: 1 .2 . 8, 1 .2 . 10, 1 .2 . 17, 1 .5. 32

in measure: 3.7. 134 --- uniformly: 1 .2 . 17 convergence inducing factor: 7 .1 . 312 convex combination: for a set Xl , . . . , Xn of vectors and a set aI , . . . , an of n n

nonnegative numbers such that L ak = 1, the vector L akxk , 3.3.

103 ---- function: 3.2. 100

k=l k=l

-�- set: in a vector space V, a subset 5 that contains every convex combination vectors in 5, 1.4. 22, 3 . 1 . 89

convexity theorem of M. Riesz: 4.9. 191 , 11 .2 . 436 convolution: 4.4. 157, 4. 10. 195, 6 .2 . 274 count ably additive: 2.2. 61 , 4 . 1 . 137 count ably subadditive: 2.2. 68 counting measure: 3.2. 97

Glossary /Index 467

covariant functor: for categories C1 and C2 , a map F : C1 H C2 of each object to an object, each morphism to a morphism, each l A to I F (A ) , and such that: when gl 0 II is defined, F (gl ) 0 F (II ) is defined and F (gd 0 F (II ) = F (gl 0 II ) , 3.5. 125

covering space, covering space triple : 10.3. 409 cover transformation (German: Decktransformation: 10.4. 418 cross ratio: 8.2. 342 curve: 2.4. 79, 4.8. 185, 5 . 1 . 204

-image: for a curve ,,(, the set "(* : 2.4. 79 --- -component : 10.2. 399 --- -connected: 10.2. 401 --- starting at yo : 5 .5 . 245 customary topology (for JR.) : 1 .2. 6 cycle: 5.5 . 249 cylinder: 1 .2 . 9

D

#(D(f))-DEFECTIVE: 9 . 1 . 326 DANIELL, P . J . : 2 . 1 . 44, 2 . 1 . 45 Daniell (DLS) functional: 2 . 1 . 45 --- measurable function: 2.2. 56 --- --- subset : 2.2.59 decomposable: 4.2. 146 defections: 9 . 1 . 369 defective function: 7.3. 330, 9 . 1 . 369 degenerate k-simplex: 1 .4. 21 degree of a map: 1 .4 . 25 DENJOY, A . : 4.7. 177 dense: 1 .2 . 1 1 DE RHAM, G . : 5.5 . 237 De Rham's Theorem: [SiT] , 5.5 . 249 derivative: 4.6. 166, 4.7. 178 diagonal: 4.5. 163 diagonalization method: 5.3. 228 diameter: of a subset S of a metric space (X, d) ,

sup { d(x, y) : {x, y} C S } , 1 .4. 22

differentiable: 4.7. 177 differential forms: 5.8. 261 dimension: of a subspace W of a vector space V, the cardinality of (any)

maximal linearly independent subset, i.e. , basis, of W, 4.7. 179 DINI , D. : 1 .2. 16 Dini's Theorem: 1.2. 16

468

DIRAC, P . A . M. : 2.2. 55 Dirac functional: 2.2 . 55 directed: 1 .2 . 10 DIRICHLET , P. G. 1.: 3.7. 133, 6.3. 285, 8.2. 342 Dirichlet region: 6.3. 285 --- 's problem: 6.3. 285 --- 's kernel: 3.7. 133 --- series: 5.9 . 269 --- 's Principle : 8.2. 342 discrete topology: 1 .2 . 6 diset : 1 .2 .10 diverge: 5.2. 205 divergence: 6.3. 287 domain of holomorphy: 12 .1 . 449 Dominated Convergence Theorem: 2 .1 . 53 dual: 3.3. 104 --- Haar measure: 1 1 .2 . 441 --- pair: 3.3. 104 --- space: 3.3. 104

Glossary /Index

dyadic rational number: for some n in N and some M in ;2;, a number of n

the form M + L EkT k , E% = Ek : 1 .2 . 16 k=l

EDGE: 1 .4. 21 EGOROV, D . F. : 2.5 . 87 Egorov's Theorem: 2.5. 87 electric charge: 6.3. 287 elliptic: 10.3. 417 --- function: 8.5. 363 endomorphism: 3.5. 117 entire: 5 .3 . 255, 7 .1 . 301

E

epimorphism: a surjective homomorphism, 2.4. 82, 3.3. 108 equicontinuous : 1 .6. 36 equivalence classes : 1 . 1 . 3 equivalence relation: for a set 5, in 52 a subset R such that:

a) {a E 5} ::::} {{a, a} E R} (R is reflexive) ; b) { {a, b} E R} ::::} { {b, a} E R} (R is symmetric) ; c) {{{a , b} E R} 1\ { {b, e} E R}} ::::} { {a, e} E R} (R is transitive) ;

1 .2 . 11 equivalent function elements: 10.2. 396

Glossary /Index

--- norms: 3.5. 117 ERDOS, P . : 5.8. 269 essentially equal: 1 .2 . 1 1 essential singularity: 5.4. 230 --- at 00: 5.6. 252 ESTERMANN, T . : 9 . 1 . 370 Euclidean metric: 1 .2. 8, 5 . 1 . 204 EULER, 1 . : 2.4. 80, 5.7. 255, 8 . 1 . 341 Euler's equations: If U (X l ' . . . ' xn ) minimizes

, then

[CoH] . 8.6. 337 ---formula: 2.4. 80 evaluation map: 3.7. 136 , 5.8. 257, 8.3. 353 evenly covered: 10.3. 409 eventually in: 1 .2 . 1 1 exponent o f convergence: 7.3. 327 --- --- divergence: 7.3. 327 exponential function: 2.4. 80 extended real number system: 1 .2 . 8 ---- JR.-valued functions: 1 .2 . 8 ---- Schwarz Reflection Principle: 8 .2 .346 exterior calculus : 5.8. 257 --- (wedge) product: 5.8. 261

FACTOR SPACE: 4.4. 153 FATOU, P . : 2.1. 53 Fatou's Lemma: 2 . 1 . 53 FEJER, 1 . : 3.7. 133 Fejer's kernel: 3.7. 133 --- Theorem: 3.7. 133 F-homotopic in A: 1 .4. 20

F

469

field: a commutative ring (q.v. ) containing a multiplicative identity with respect to which each element is invertible, 1 . 1 . 3

filter: 1 .5. 32 --- base: 1 .5 . 32 --- corresponding to a net : 1 .5 . 32

470 Glossary /Index

--- generated by a filter base, 1 .5 . 28 finer (filter) : 1 .5 . 32 finite cylinder: 1 .2. 9

intersection property: 1 .2 . 12 --- measure space: 4.2. 146 --- subset (of I,p(X) ) : a subset S such that each finite set of elements of S has a nonempty intersection, 1 .2. 13

finitely additive set function: for a ring 5 of sets, a map

¢ : 5 '3 E H ¢( E) E ffi. such that if En , 1 � n � N are pairwise disjoint elements of 5, then

¢ (u :=1 An) = t, ¢ (En) , 2.2 . 61

first (Baire) category: 1 .3. 18 FISCHER, E . : 3.7. 135 Fischer-Riesz Theorem: 3.7. 135 fixed point property: 1 .4. 26 form(s) (O-form, I-form, 2-form) : 5.8. 258 FOURIER, J . : 3.7. 1 15, 4.9. 169, 4.9. 172 Fourier coefficients: 3.7. 132

series : for a function f having the sequence {cn }nEZ of Fourier coefficients, the series L cne2n7rit , 4.10. 192

nEZ --- transform: 4.9. 189 FRAENKEL, A . : 1 . 1 . 3, 4.5. 160 frequently: 1 .2. 1 1 Fubinate: 4.4. 153 FUBINI, G . : 4.4. 153 Fubini's Theorem: 4.4. 155 functional calculus : a calculus for defining functions of a finite set of ele

ments of [SJ]c ' 3.7. 131 function element : a pair (I, Q) consisting of a region Q and an f in H (Q) :

10. 1 . 382 --- lattice: 2 . 1 . 44 ---- pair: 10.2. 396 functor: v. category fundamental group: 5 .5 . 245, 10.6. 430 --- set: 8.4. 362 Fundamental Theorem of Algebra (FTA) : 9 . 1 . 369 Fundamental Theorem of Calculus (FTC) : 4.6. 164

G

Glossary /Index

GAP, NONGAP: 10. 1 . 388 GAUSS, C. F . : 5 .9 . 269 r-automorphic: 8.4. 361 GELFAND, I . M. : 3.5. 124-125, 4.9. 189 Gelfand-Fourier transform: 4.9. 189

map: 3.5. 125 --- -Mazur: 3.5. 124 generated (a-ring) : 2.2 . 58 genus : 7.2. 318 germ: 10.2. 391 global Cauchy integral formula: 5.4. 235

471

globally injective: of a function in H (fl) , that it is injective on fl, 5.3. 223 gradient : 6 .3. 287 GRAM, J. P . : 3.2. 98 Gram-Schmidt process: 3.2. 98 graph: 3.3. 109 Great Picard Theorem: 9. 1 . 369, 9.4. 376 GREEN, G . : 5.8. 262, 10.4. 372 Green's formula: 5.8. 2.62 --- function: 8.5. 363, 10.3. 417 ---- Theorem: 5.8. 262 group: a set G and a map · : G2 '3 {g, h} H g . h E G such that · is associa

tive; G is assumed to contain an identity e such that e . g == g; for each g there is an h such that h . g = e, 2.5. 84

--- algebra: 4.9. 189 GRUNSKY, H . : 9.4. 375 GUTZMER, A . : 5.3. 218 Gutzmer's coefficient estimate: 5.3. 218

HAAR. A . : 4.9. 187 Haar measure: 4.9. 187

H

HADAMARD, J . : 7.3. 328, 10. 1 . 388, 11 .2 . 435 Hadamard's Gap Theorem: 10 .1 . 388 --- Three-circles Theorem: 1 1 .2 . 435 --- Three-lines Theorem: 1 1 .2. 435 HAHN, H . : 4. 1 . 139 Hahn-Banach Theorem: 3.3. 105 Hahn decomposition: 4. 1 . 140 half-line: for two points x and y in a vector space, the set

{ x + ty : 0 � t < oo } , 1 .4. 26

half-open: 1 . 1 . 5

472

--- cube vertexed at (k, m) : 4.6. 165 --- interval: l . l . 5, 4.5. 163 --- n-dimensional cube: 1 . 1 . 5 --- n-dimensional interval: 1 . 1 . 5 HAMEL, G : 3.3. 104 Hamel basis: 3.3. 104 HARDY, G . H . : 7.3. 334 Hardy spaces: 7.3. 334 harmonic: 5.3. 227

Glossary /Index

--- analysis on locally compact abelian groups: 4.9. 192 ---- conjugate: 5.3. 227 --- measure: 6 .5 . 300 HARNACK, A . : 6.2. 281 Harnack's Theorem: 6.2. 281 --- inequality: 6.5. 280 HARTOGS, F . : 6.3. 257, 12 .1 394 Hartogs's Theorems: 6.3. 294 HAUSDORFF, F . : 1 .2 . 8, 1 .5 . 32, 1 .6 . 34, 4.9. 187, 11 .2 . 442 Hausdorff Maximality Principle: Every nonempty poset contains a maximal

ordered subset, v .Zorn's Lemma. 1 .5 . 32 --- space: 1 .2. 8 --- topology: 1 .2 . 8 --- uniformity: 1 .6 . 34 ---/Young and F . Riesz Theorem: 1 1 .2 . 442 hereditary: 2.2. 68

a-ring: 2.2. 68 HILBERT, D . : 3.2. 82, 3.5. 1 10, 8.6.337 Hilbert space: 3.2. 97, 3.5. 1 17 HOLDER, 0. : 3.2. 77 Holder's inequality: 3.2. 93 --- --- extended: 3.7. 135 holomorphic: 5.3. 185 --- autojection: 8 . 1 . 341 --- function: 5.3. 209 --- near x: 10.2. 404 --- on AS: 10.2. 406 holomorphically compatible: 10.2. 403 homeomorphic: of two topological spaces X and Y, that there is a homeo-

morphism X H Y, 1 .4. 22 homeomorphism: 1 .2 . 7 homologous: 5.5.249 homology: 5.5 245 homotopic, homotopy (in A) : 5.5. 245 H (U)-hull: 12 .1 . 449 HURWITZ. A . : 5.4. 243

Glossary /Index

Hurwitz's Theorem: 5.4. 243 hyperbolic: 10.3. 417

473

hyperplane: in a vector space, the translate of the span of a finite number of vectors, 1.4. 22

��- of dimension n: the translate of the span of n linearly independent vectors, 4.7. 178

I

IDEAL: 3.5. 124 Identity Theorem: 5.3. 255 identity modulo a left (right) ideal: 3.5. 124 image: 2.4. 79, 3.6. 129, 4.8. 145, 5.6. 253 immediate analytic continuation: 10. 1 . 382 index of a curve with respect to a point : 2.4. 83

subgroup H in a group G: the cardinality of the set { gH : g E G } , of cosets of H, 4.5. 159

indivisible: 2.5. 84 induced by T: 1 .2. 6

inner measure: 2.2. 68 ---- outer measure: 2.2. 68 �-� topology: 1 .2. 12 infinite-dimensional: of a vector space V, that for each n in N there are n

linearly independent vectors in V, 3.3. 104 infinitely differentiable: of a function f in ffi.1R that for each n in N, f(n)

exists, 2.4. 80 infinite product : 7. 1 . 311 , 7.2. 313 infinity: 6.3. 287 injection, injective: 1 .2 . 7 inner measure: 2.2. 68 ��- product : 3.2. 96

--� regular: 2.3. 78 integers: 1 . 1 . 3 integrable: 2.2. 63 integration by parts: 4. 10. 193 interior: 1 . 1 . 5, 1 .2 . 7 intermediate value property: of a function f in ffi.IR , that if a -j. b and � is

between f(a) and f(b) , for some c between a and b, f(c) = � , 1 .7. 43 ��- �� theorem for derivatives: If f E ffi.1R and f is a derivative,

then f enjoys the intermediate value property, q.v . , 4.6. 176 interval: 1 . 1 . 5 inversion: 8.2. 344 irregular point : 10.2. 395 isolated point : 1 .2 . 10

474 Glossary /Index

--- essential singularity: 5.4. 230 --- singularity: 5.4. 230 isometric: of a map f : X H Y between metric spaces (X, d) and (Y, J) ,

that J[J(a) , f(b)] == d(a, b), 8.4 . 362 --- circle: 8.4. 359 isometrically isomorphic: 3.2. 97, 3.6. 130 isometry: an isometric map, 3.4. 112

J

JACOBI , C . G . J . : 4.7. 182 Jacobian matrix: for a differentiable map

h . 8 (YI , . . . , Yrn ) 8y] t e m x n matnx in which the ij entry is - , 4.7. 182 8 (X I , . . . , Xn ) 8x,

JENSEN, J . 1 . W. V . : 3.2. 102, 7.3. 332, 1 1 . 1 . 431 , 12 .1 . 447 Jensen's formula: 7.3. 332 --- inequality: 3.2. 102, 1 1 . 1 . 431 , 12. 1 . 447 --- Lemma: 7.2. 321 JORDAN, C . : 2.5. 83, 4. 1 . 142, 5.7. 255 , 7. 1 . 305 Jordan curve: a homeomorphism "/ : 1I' '3 t H "/(t) E C, 5.8. 234

Curve Theorem: If f : 1I' H ffi.2 is a homeomorphism, ffi.2 \ f (1I') consists of two (open) components, one bounded, and one unbounded, and of which f(1I') is the common boundary. 2.5. 83, 7. 1 . 305

--- decomposition: 4.2. 142 --- inequality: 5.7. 255 jump function: 4. 10. 196

KERNEL: 3.6. 128 K-fold Fubinate: 4.4. 154 K-fold pre-Fubinate: 4.4. 154 KOEBE, P . : 8 .1 . 342 KONIG, R.: 8 .1 . 342 KRONECKER, 1 . : 3.3. 108 Kronecker's function: 3.3. 108

00

K

L

LACUNARY: of a series L Cn , that for some positive E, and a sequence n=O

{PdkEN ' PHI 2: 1 + E, k E N, and if Pk < n < Pk+l , then Cn = 0 (the Pk series has gaps) , 1 1 .2. 443

Glossary /Index

LANDAU , E . : 9 . 1 . 370, 9 .2 . 371 , 9.4. 377 Landau's Theorem: 9.4. 377 LAPLACE, P . S . DE: 5.3. 227 Laplacian: 5.3 . 227 last maximal term: 7.3. 324 lattice: 2 . 1 . 44 LAURENT, P. A . : 5.4. 237 Laurent series: 5.4. 237 LEBESGUE, H . : 1 .2 . 15, 2 . 1 . 52, 2 .2 . 67, 2.4. 78, 4 .6 . 168 Lebesgue integrable: 2.4. 78

integral: 2.4. 78 --- measurable: 2.2 . 67 --- --- function: 2.4 . 78 --- --- set: 2.2 . 67 --- measure: 2.4. 78 --- numbers: 1 .2 . 15 --- 's Covering Lemma: 1 .2 . 15 --- 's decomposition: 4.2. 144 ---- set: 4.6. 168 ---'s decomposition: 4.2. 144 ---'s Monotone Convergence Theorem: 2 . 1 . 52 left adverse: 3.5. 118 --- a-translate: 2.5 . 84 --- inverse: 3.5. 117 --- -closed interval: 1 . 1 . 5 --- -continuous : 4.10. 196

475

--- -hand derivative: of an f in ffi.IR , lim f(b) - f(x) , when it exists, btx b - x

3.2. 102 length: 4.8. 185 lift : 10.3. 409 lim-closed: 2.2. 57 lim -closed: 2.2 . 57 lim-closed: 2.2. 57 limit point: 1 .2 . 10 LINDELOF, E . : 1 1 .2 . 444 linear and convex programming: [ZAJ , 11 . 1 . 431

functional: for a vector space V, a map ¢ : V H C such that ¢(ax + by) = a¢(x) + b¢(y) , 3.3. 104

--- combination: for a set S of vectors,

476 Glossary /Index

v.span linearly dependent : not linearly independent , q.v.

independent : of a set {x>,} ),EA of vectors, that for each finite subset

a of 1\, if L a),x), = 0, then a), I " = 0, 1 .4. 21 ),E "

LIOUVILLE, J . : 3.5. 122, 5 .3 . 218 Liouville's Theorem: 3.5 . 122, 5.3. 218 LIPSCHITZ, R. : 4 .10 . 192 Lipschitz function: 3.2. 96, 4.10. 172 Lipschitian: of an f in JR.(a,b) , that for some Lipschitz function (q.v. ) L(x, y)

and some positive D , if a < x < y < b, then

I f(x) - f(y) 1 � L(x, Y) lx - Y i n , 3.2. 101

Little Picard Theorem: 9.3. 374 lives : 4.2. 143 local inverse: 10.1 . 384 locally compact : 1 .2 . 12 --- connected: 1 .2 . 12 --- convex topqlogical vector space: 3 .1 . 89 --- curve connected: 1 .2 . 12, 10.2 . 406 --- injective: of a map f, that it is injective in some neighborhood of

each point , 5.3. 223 --- representable: 5 .1 . 204 --- simply connected: 10.2. 402 local uniformizer: 10.2. 396 logarithmic function: 2.4. 80 logoid derivative: 8.2. 347 loop: 5.5. 246 --- homotopic: 5.5. 246 --- homotopy: 5.5. 246 lower semicontinuous : 1 .2. 9

MACLAURIN, C . : 2.4. 79

M

Maclaurin polynomial: for an f in eN (JR., JR.) when 0 � k � N,

N f(n) (o) L __ ,_xn , 2.4. 79 n .

n=O

--- series: for an f in eOO(JR., JR.) , 00 f(n) (0) '" __ xn , 2.4. 79 L-, n ! n=O

Glossary /Index

majorization: 5.3. 223 majorize: 5.3. 223 maximal set of function elements : 10.2. 391 maximal simply connected subregion: 5.9. 265 Maximum Modulus Theorem: 5.3. 220 ---------- in Coo : 5.6. 253

- - - Principle: 6 .1 . 271 ------- in D relative to B: 1 1 . 1 . 432 MAZUR, S . : 3.5. 124 Mean Value Property: 6 . 1 . 271 measurable cover: 2.2. 68 --- kernel: 2.2. 68

477

---- partition: of a set E in a a-ring 5 , in 5 , a countable subset {En} nE]\/ such that E = U En, 4. 1 . 138 nE]\/

measure: 2.2. 6 1 space: 2.2. 61

meromorphic: 5 .4 . 231 , 10 .5 . 422 ---- in an open set: 5.4. 231 Metric Density Lemma: 4. 10. 192

space: 1 .2. 8 mid-closed: 2.2. 56 midpoint convex: 6.3. 284 MINDA, C. D . : 9 . 1 . 370 Minimum Modulus Theorem: 5.6. 253 MINKOWSKI, H . : 3.4. 114 Minkowski functional: 3.4 . 114 ---'s inequality: 3.2. 93 MITTAG-LEFFLER, G . : 7. 1 . 301 Mittag-Leffler's Theorem: 7.1. 301 MOBIUS, A. F. : 8.2. 343 Mobius-invariant : 8.2. 348

strip: 10.2. 365 --- transformation: 8.2. 342 modular: 3.5. 124

group: 8.4. 357 monodromy: 10.4. 420 Monodromy Theorem: 10.4. 420 monotone: of a sequence {an} nE]\/' that for each n, an :::; an+l or an 2': an+l :

1 .2 . 9; of a class C of functions, that together with every monotonely increasing or monotonely decreasing sequence {In} nE]\/ contained in C, the function lim In i s in C, 2.2. 57 n---+oo

--- ring of sets: 2.5. 85

478

monotonely increasing function: in ffi.1R an f such that

{a < b} '* {J(a) ::::: f(b) } , 1 .2 . 13

MONTEL, P . : 5.9. 267, 9.4. 335 Montel's Theorem: 9.4. 377 MORERA, G . : 5.3. 224 Morera's Theorem: 5.3. 224 MROWKA, S . : 1 .7. 42 f.-l* -a-finite: 2.2. 69

Glossary /Index

multiplicative linear functional: for an algebra A, in A', an element x' such that x'(x · y) = x'(x) . x' (y) , 3.5. 124

multiplicatively convex function: 11 .2. 437 multivalued function: for a set X, in X2 , a subset S such that for each x

in X there is in S a subset Sx contained in {x} x X, 10.2. 392 multivaluedness: 10.2. 391 mutually singular: 4.2. 143

N

NATURAL BOUNDARY: the boundary of a region in which a function f is holomorphic and incapable of extension to a function holomorphic in a proper superregion, 6.2. 279, 7. 1 . 312, 10.2.406

--- numbers: 1 . 1 . 3 n-dimensional complex analytic manifold: 10.2. 403 --- Lebesgue measure: 4.5. 162 n + I-cell: 1.4. 21 negative set: 4.1. 138 neighborhood: 1 .2. 7, 5.6 . 252, 10.2. 397 net: 1 .2 . 10 --- corresponding to the filter: 1 .5. 32 NEUMANN, C . : 8 .1 . 342 NEUMANN, J . VON: 4.2. 143, 1 1 . 1 . 431 NEVANLlNNA, R. : 7.3. 323 nicely shrinking: of a sequence {Ern}"nEN of sets in ffi.n , that for some fixed

x, each Ern is contained in some B (x, rrnt and for an m-free constant a(x) , >. (Ern ) � a(x)>. [B (x, rrntJ , 4.6. 169

NIKODYM, 0.: 4.2. 144 nonisolated singularity: 5.4. 232 nonnegative count ably additive set function: 2.2. 61 --- linear functional: 2 .1 . 45 nonorientable: 10.2. 403 nonsingular: of a linear map T between finite-dimensional vector spaces,

that T- 1 (O) = {O } , 4 .7. 178 norm: 1 . 1 . 5, 3. 1 . 89

Glossary /Index

normable: 3.7. 136 normal element of [SJlc : 3.6. 130

family: 1 .6 . 38

479

--- subgroup: in a group G, a subgroup H such that for all 9 in G, g- 1 Hg = H, 8.4. 358

normalized function of bounded variation: 4.10. 197 norm-bicontinuous: of a bijection between normed vector spaces, that it is

bicontinuous with respect to the topologies induced by the norms, 3.2. 98

norm-decreasing: of a map T between normed vector spaces, that

I IT(x) 1 1 � I lx l l , 3.5. 123

normed ring: v.Banach algebra, 3.5. 1 16 space: 3 .1 . 89

norm-induced: of a topology in a I I I I -normed vector space, that

{ x : I lx l l < E, E > 0 }

is a base of neighborhoods at 0, 3.4. 1 1 1 --- -separable: of a norm-induced topology, that it is separable, q.v . ,

3.2. 98 nowhere dense: 1.3. 18 n-simplex: 1.4. 21 null function: 2.2. 62

homotopic: 5.5. 245 �--- set: 2.2. 62

I-FORM: 5.8. 258

o

one-parameter family of graphs: 5.5. 245 -point compactification: 1. 7. 42, 5 .6 . 252

open: 1 . 1 . 5, 1 .2. 6 , 1 .2 . 6 --- annular sector: 5.4. 234 --- ball: 1 .2 . 8 --- disc: 5 .1 . 205 -��- interval: 1 .2. 6 Open Mapping Theorem: 3.3. 108, 5.3. 221 operator: 3.6. 128 orbit: 10.3. 414 order of a function: 7.3. 324

-��- pole: 5.4. 231 --- --- growth: 7.3. 324

480 Glossary /Index

ordered: of a subset of a poset, that any two elements of S are -<-related: 1 . 1 3

ordinal number: an equivalence class of well-ordered sets, 1 . 1 . 5 orientable: 10.2. 403 oriented boundary: 7.1 . 302 --- complex interval: 1 . 1 . 5 --- cycle: 7. 1 . 304 --- real interval: 1 . 1 . 4 orthogonal, orthonormal: 3.2. 96 OSTROWSKI , A. : 10. 1 . 387 Ostrowski's Theorem: 10. 1 . 387 outer measure (induced) : 2.2. 68 --- regular: 2.4. 78 overconvergence: 10. 1 . 388

PARABOLIC: 10.3. 417

p

parameter of regularity: for a subset S of ffi.n , and

Rs �_ef { R .. R b d S R } a cu e an C ,

An (S) sup �( )

; [Sak] , 4.6. 169 RE Rs An R

parametric equations: 10.2. 395 PARSEVAL( -DESCHENES ) , M . A . : 3.2. 97 Parseval's equation: 3.2. 97 partially ordered set: 1 .2 . 10 partition: 2.2. 73 --- of unity: 1 .2. 16 patch: 10.2. 406 period: for a group G, a set S, and an J in SG , in G an element p not the

identity of G and such that J(px) == J(x), 2.4. 80 --- parallelogram: 8.6. 366 periodic: for a group G, a set S, and an element p not the identity of G, of

an J in SG , that J(px) == J(x), 3.7. 133 Permanence of Functional Equations : 10. 1 . 390 perpendicular: 3.2. 96 PERRON , 0. : 6.3. 252-254 Perron family: 6.3. 288 --- J-family: 6.3. 290 --- function: 6.3. 290 p-face: 1.4. 21 PHRAGMEN, E . : 11 .2 . 389, 1 1 .2 . 444 Phragmen-Lindelof Theorem: 11 .2 . 444

Glossary /Index 481

PICARD, E . : 9 .1 . 369, 9.4. 376 piecewise P: of an f in ffi.IR , that off a finite set S, f enjoys the property P,

5.2. 206 �� differentiable: 4.8. 186 �� linear: 5.2. 206 �� well-behaved: 4.8. 186 PLANCHEREL, M . : 1 1 .2 . 441 Plancherel's Theorem: 11 .2 . 441 POINCARE, H . : 8.6. 361 , 10.2. 394 Poincare's Theorems: 8.4. 361, 10.2. 394 point function: 4.6. 164 POISSON, S. D . : 6.2. 273, 6.2. 275, 6.2. 277, 6.2. 244, 6.3. 249 Poisson-Jensen formula: 6.2 . 277 Poisson kernel: 6.2 . 273 �� modification: 6.3. 286 �� 's formula: 6.2 . 275 �� transform: 6.2 . 275 polarization: 3.6. 129 pole: 5.4. 230 �� at 00: 5.6. 252 �� branched: 10.2. 399 �� of order n: 5.4. 232, 10.2. 399 polydisc: 10.1 . 390 polygon: a union n of finitely many complex intervals { [ak , ak+dL<k<n- l ;

for some curve ,,( , n = "(* : 1 .7. 39 - -polygonally connected: 1 .7. 39 polynomial function: for some N in N, a map

f : en '3 Z H L co:zO: , 3.2. 99 lo: l :S:N

Polynomial Runge: 7. 1 . 306 , 12. 1 . 448 polynomially convex: 12 .1 . 450 polyregion: 12. 1 . 466 POMPEIU, D . : 5.8. 263 Pompeiu's Theorem: 5.8. 263 PONTRJAGIN, 1 . : 4.9. 189, 1 1 .2 . 440 Pontrjagin's Duality Theorem: 4.9. 189, 11 .2 . 440 poset: 1 .2. 10 positive definite: for a vector space V, of a map

( , ) : V2 '3 {x, y} H (x, y) E e,

that (x, x) 2: 0 and (x, x) = 0 iff x = 0, 3.6. 128

482 Glossary /Index

--- degree: of a polynomial L co:zO: , that min { In l } > 0, 5.6. 252 lo: l �N

--- set: 4.1 . 138 potential function: for a vector field E, a scalar function <1> such that

E = grad <1>, 6.3. 287 precompact : of a set S in a topological space, that the closure S is compact ,

1 .6 . 38 pre-Fubinate: 4.4. 134 preserves angles: 8 . 1 . 336 , 8 . 1 . 339 Prime Number Theorem: [Schap] , 5 .9. 269 principal part: 5.4. 239 Principle of the Argument: 5.4. 242 PRINGSHEIM, A . : 10.1 . 384 Pringsheim's Theorem: 10. 1 . 384 probability theory: [Kol] , 2.2. 66, 1 1 . 1 . 431 product: 3.5. 1 16 --- curve: 5.5. 245 --- measure: 4.4. 151 --- rule: 4.7. 184 --- topology: 1 .2 . 9 projection: 1 .2 . 10, 3.6. 129, 10.2. 391 properly discontinuous : 8.4. 358 punctured disc: 5.4. 234 P-variation: 4.6. 169

Q QUOTIENT ALGEBRA: for an algebra A and an algebra homomorphism ¢,

the image ¢(A) regarded as an algebra by virtue of ¢; ¢(A) is the quotient Aj¢-l (O) , 3.5. 124

--- norm: 3.5. 124 --- set: 2.2. 62 --- space: for a vector space X and a subspace Y, { x + Y x E X },

3.5. 123

RADIUS OF CONVERGENCE: 5.2. 205 RADON, J . : 4.2. 144 Radon-Nikodym derivative: 4.2. 144 --- Theorem: 4.2. 144

R

range: for a map f : X H Y, f(X) , 1.2. 21 rational numbers: 1 . 1 . 3 re-arrangement of series: 10. 1 . 384 rectifiable: 2.4. 79, 4.8. 185

Glossary /Index

recursively: [Me] , 5.3. 223 refinement : 1 .5 . 32 refines : 1 .5 . 32 reflection: 8.2. 344 reflexive: 3.4. 112 region: 1 . 1 . 3, 1 .2 . 1 1 , 5 . 1 . 204 regular ideal: 3.5 .124 --- measure (space) : 2.3. 78

point : 10. 1 . 384, 10.2. 399 relative topology: 1 .2. 6 relatively closed, compact , open: 1 .2 . 17 remainder formulre: 2 .4 . 80 removable singularity: 5.4. 230 reproducing kernel: 8.3. 353 residue: 5.4. 239 Residue Theorem: 5.4. 239 restriction: 1 .2 . 1 1 reversed inclusion: 1 . 2 . 10 RrcKART, C . E . : 3.5. 121 RIEMANN, B.: 1 .2. 12, 2.2 . 73, 4.6. 170, 4.8. 185, 5.6. 251 , 10.2. 405 Riemann integrable: 2.2. 73

partition: 2.2. 73 --- 's Mapping Theorem: 8 .1 . 336 --- sphere: 5.6. 251 --- -Stieltjes integral: 4.6. 170 ---'s zeta function ( : 5.9. 269 --- -Stieltjes sum: 4.6. 170 --- surface: 1 .2 . 12, 10.2. 405 RIEsz, F . : 2.3. 76 , 3.6. 128, 3.7. 135, 4.3. 150, 1 1 .2 . 440, 11 .2 . 442 F. Riesz's Theorems: 2.3. 76, 3.6. 128, 4.3. 150, 1 1 .2 . 440 RIEsz, M . : 4.9. 191 , 1 1 .2. 436 M. Riesz's Convexity Theorem: 4.9. 191 , 11 .2 . 436 right adverse: 3.5. 118 --- -open: 1 . 1 . 5

483

h d d · . C f · IDlIR 1· f(b) - f(x) h . . ---- an envatlve: lor an In m. , 1m , w en It eXIsts, x<btx b - x

3.2. 102 --- ideal: 3.5. 124 --- inverse: 3.5. 117 --- open interval: 1 . 1 . 5 ring: a set R and two maps

+ : R2 '3 {a, b} H a + b E R; . : R2 '3 {a, b} H a . b E R;

484 Glossary /Index

both + and · are associative; + is commutative; . is +-distributive, 1 . 1 3

--- of sets: 2.2. 58 rotation: in [lR.n l e ' a T such that I IT(x) II == I lx l l (T i s a linear isometry) , 4.5

162 --- -invariant : 4.5. 162 ROUCHE, E . : 5.5. 244 Rouche's Theorem: 5.4. 244 RUNGE, C . : 7. 1 . 301 , 7. 1 . 305-307, 12 . 1 . 450 Runge domain: 12. 1 . 450 --- 's Theorem: 7. 1 . 305

variant : 7. 1 . 307 Running Water Lemma: 4.6. 172 IR-valued functions : 1 .2 . 8 JR.-vector space: 3 .1 . 89

SAKS, S . : 4.6. 169 saturated: 10.2. 403 SCHMIDT, E . : 3.2. 98 SCHOTTKY, F . : 9. 1 . 370, 9.4. 376 Schottky's Theorem: 9.4. 376

s

SCHWARZ, H . A . : 3.2. 96 , 3.7. 135, 6 .2 . 280 Schwarzian derivative: 8.2. 347 Schwarz inequality: 3.2. 96 --- Reflection Principle: 6.2 . 278 ---'s formula: 6.2 . 280 ----'s Lemma: 7.2. 322, 12. 1 . 447 second (Baire) category: 1 .3. 18 second countable: 1 .2. 7 SELBERG, A . : 5.9 . 269 self-adjoint: 3.6. 130 semihomogeneous : 2 .1 . 47 seminorm: 3.3. 105 separable: 1 .2. 7 separated: 1 .2. 1 1 separating algebra: 3.5. 126 --- elements in a dual space: 3.3. 108 shared: 7. 1 . 303 sheaf: 10.2. 393 sheets: 10.2. 395 shrinks nicely: [Rud] , 4.6. 169 sides: 7.1 . 302 a-algebra: 2.2. 58

Glossary /Index 485

--- -compact : of a subset 5 of a topological space, that 5 is the union of count ably many compact sets, 4.2. 146

--- -finite: 4.2. 146 --- -ring: 2.2. 58 signed measure (space) : 4 .1 . 138 simple function: 2.2. 63

pole: 5.4. 241 --- zero: 5.4. 241 simplex: 1 .4. 21 simply connected: 5.5. 246, 10.2. 401 singleton: of a set 5, the #(5) = 1 , 2.2 . 67 singular: of a linear map T between finite-dimensional vector spaces, that

T- 1 (O) -=j:. {O}, 4.7. 178 --- component : 4.2. 144 --- point : 5.4. 232, 10 . 1 . 384 singularity: 5.4. 230 SORGENFREY, R. H . : 1 .2 . 6 somewhere dense: of a subset 5 of a topological space X, that for some

neighborhood N, 5 n N ::) N: 3.3. 109 Sorgenfrey topology: 1 .2 . 6 span: (noun ) : for a subset 5 of a vector space, the set of all (finite) linear

combinations of elements in 5: 1.4. 22 --- ( verb) : a subset 5 of a vector space V spans V iff span (5) = V:

3.2. 97 Spectral Theorem: 3.6. 130 spectral radius: 3.5. 123 spectrum of an algebra: 3.5. 124 ---- ---- --- element x: 3.5. 120 --- --- --- operator in [SJ] c : 3.6. 131 SPERNER, E . : 1 .4. 27 Sperner map: 1 .4. 27 --- simplex: 1 .4. 29 ---'s Lemma: 1 .4. 27 stalk: 10.2. 394 standard basis: for en , the set {ei �f (ei 1 , . . . , ein )} . of vectors such l �"�n

that eij = Jij , 4.7. 182 star-shaped: 5.3. 226 STEINHAUS , H. : 3.3. 110 step-function: an ffi.- linear combination of characteristic functions of half-

open intervals of the form [a, b) : 2.2. 75 stereo graphic projection: 5.6. 252 STIELTJES, T. J . : 4.6. 149 Stieltjes integral: 4.6. 49 STOKES, G . G . : 5.8. 235

486

Stokes's Theorem: [Lan,Spi] , 5.8. 233 STONE, M. H . : 2.2. 64, 3.5. 126, 3.7. 136 Stone-Cech compactification: 3.7. 136 Stone's Theorem: 2.2. 64 Stone-WeierstraB Theorem: 3.5. 126 straight line through a: 8.1 . 338 strictly separating: 3.5. 126 strong form of FTA: 5.4. 244 stronger topology: 1 .2. 6 strongest topology: 1 .2. 6

Glossary /Index

subadditive: of a functional !, that !(I + g) � !(I) + !(g ) , 2.2. 68 subharmonic: 6.3. 285, 10.2. 404, 1 1 . 1 . 432 --- in Q: 6.3. 285 --- --- the wide sense: 6.3. 293 subordinate: 1 .2. 16 subspace: in a vector space V, a subset W that is also a vector space, 3.3.

105 summable: 3.3. 1 1 1 superadditive: 2 .1 . 48; 4 . 1 . 141 super harmonic: 6.3. 287, 10.2. 404 support : 1 .2. 15 supporting line: for the graph G of y = ¢( x) and a point P �f (p, f (p) ) on

G, a line L through P and such that near P, L is below G; 3.2. 102 surjection: 1 .2 . 7 surjective: 1 .2. 7 SZEGO, G . : 8.3. 354 Szego's kernel: 8.3. 354

T

THE CANTOR SET: 1 .2 . 13 theory of games : [NeuM] , 1 1 . 1 . 431 thick: for a measure space (X, S , f.-l) , of a subset Y such that for every E in

S, f.-l* (E \ Y) = 0, 4.5. 160 TIETZE, H . : 1 .7. 41 Tietze's Extension Theorem: 1 .7. 41 THORIN, G . 0. : 11 . 1 . 431 , 1 1 .2 . 434 Thorin's Theorem: 11 . 1 . 431 TONELLI, 1 . : 4.4. 156 Tonelli's Theorem: 4.4. 156 topological algebra: 6.4. 296

Glossary /Index

--- field: a field OC in which the maps

OC2 '3 {a, b} H a + b, OC2 '3 {a, b} H ab,

OC \ {O} '3 a H a- I ,

are continuous with respect to the topology of OC, 6.4. 296 --- group: 1 .6. 34, 4.9. 187 --- space: 1 .2. 6

vector space: 3 .1 . 89 topologist's sine curve: 10.2. 402 topology: 1 .2 . 6 total variation: 4.6. 169

487

totally disconnected: of a set S in a topological space, that the only com-ponents of S are points, 10.3. 414

--- finite: 2.5. 85, 4.2. 144 ----- a-finite: 2.5. 86, 4.2. 146 transcendental: 9 .1 . 369 transition map: 10.2. 402 transitive: of a partial order --<, that

{{a --< b} !\ {b --< e}} ::::} {a --< e} , 1.2. 10

translate: 2.5. 84, 4.5. 163, 9.2. 371 translation: 4.5. 162

-invariant : 2.4. 78, 4.5. 159 --- /scaling: 3.3. 94 triangle inequality: 3.2. 96 trigonometric functions : 2.4. 79 trivial topology: 1 .2. 6 TYCHONOV, A . : 1 .2 . 13 Tychonov's Theorem: 1.2. 13 type: 7.3. 324

u

U-INDUCED NEIGHBORHOOD: 1 .6 . 35 U-induced uniform topology: 1 .6 . 35 ultrafilter: 1 .5 . 32 unbounded near zero: 5.4. 230 uniform space: 1 .6 . 34

topology: 3.4. 1 1 1 uniformity: 1 . 6 . 34 uniformization: 10.2. 398 Uniformization Theorem: 10.5. 422

488

uniformly continuous : 1 .6 . 35 unit circle: 2.4. 81 unitary: 3.6. 131 univalent: of a map, that it is injective, 8.1. 336 universal covering space: 10.3. 411 unshared: 7. 1 . 303 upper semicontinuous: 1 .2 . 9 URYSOHN, P . : 1 .2 . 15 Urysohn's Lemma: 1 .2. 15

VALUES : 10.2. 392 v

vector: a map r '3 "I H x"! E X: 1 . 1 . 5 --- space: an abelian group V over a field OC: 2 .1 . 44 --- --- homomorphism: 3.3. 103 vertex: 1.4. 21 , 4.8. 186, 7. 1 . 304 vicinity: 1 .6 . 34 VITALI, G . : 2.3. 78, 4.6. 169, 5.3. 206 Vitali-Caratheodory (Theorem) : 2.3. 78

Glossary /Index

Vitali's Covering Theorem: If I is a set of n-dimensional intervals, and for each positive E and each x in a set E, there is in I an I such that x E I and )..n(I) < E, then if 15 > 0 and )"� (E) < 00, for some finite set

11 , . . . , !rn of pairwise disjoint elements of I, )..� ( E \ U ;=Jj) < 15. 4.6. 169 --- Theorem: 5.3. 228

WEAK, WEAK': 3.4. 112 weaker, weakest : 1 .2. 6 wedge product : 5.8. 261

w

WEIERSTRASS, K . : 3.5. 126, 5.4. 230, 7. 1 . 310, 7. 1 . 312, 8.6. 366 WeierstraB Approximation Theorem, 3.5. 126

elliptic function: 8.6. 366 --- product representation: 7. 1 . 312 --- WeierstraB-Casorati Theorem: 5.4. 230 well-behaved: 4.8. 186 --- -ordered: of an ordered set X, that each subset has a unique min

imal element , 1 . 1 . 5 Well-ordering Axiom: Every set X may be well-ordered, i.e. , there is an

order --< such that for any two elements x and y of X, x --< y or y --< x and for any subset Z of X, there is in Z a (unique) z such that for any other element z' of Z, z --< z' (every subset has a least element) . 1 .5 . 32

Glossary /Index

WEYL, H . : 10.2. 395 winding number: 5.2. 208 W-structure: 10.2. 391

x

x-FREE: independent (free) of x, 1 .6 . 37 X-holomorphic, X-meromorphic: 10.2. 404 x-neighborhood: 1 .7. 39 x-section: 4.4. 153

y

YOUNG, G . C . : 4.6. 177 YOUNG, W . H . : 4.�. 190, 1 1 .2 . 386 , 1 1 .2 . 442

ZAREMBA, S . : 8 .1 . 342 ZERMELO, E . : 4.5. 160

z

Zermelo-F'raenkel system of axioms: [Me] , 4.5 160 zero: 5.4. 231

of order or multiplicity n o : 5.4. 232 ZORN, M . : 1 .2. 13, 1.5. 32

489

Zorn's Lemma: In a poset (r , --<) , if every ordered subset has an upper bound in r, for each "I in r, there is in r a maximal element f.-l such that "I --< f.-l, 1 .5 . 32

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