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TOOLS OF THE TRADE

Introduction

to

Advanced

Mathematics

Paul J. Sally, Jr.

AMERICAN MATHEMATICAL SOCIETY

TOOLS OF THE TRADE

Introduction

to

Advanced

Mathematics

TOOLS OF THE TRADE

Introduction

to

Advanced

Mathematics

Paul J. Sally, Jr.

Providence, Rhode Island

http://dx.doi.org/10.1090/mbk/055

2000 Mathematics Subject Classification. Primary 26–01;Secondary 26A03, 26A06, 12J25.

For additional information and updates on this book, visitwww.ams.org/bookpages/mbk-55

Library of Congress Cataloging-in-Publication Data

Sally, Paul.Tools of the trade : an introduction to advanced mathematics / Paul J. Sally, Jr.

p. cm.Includes index.ISBN 978-0-8218-4634-6 (alk. paper)1. Mathematics—Textbooks. I. Title.

QA37.3.S25 2008510—dc22 2008024594

Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.

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10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08

This book is dedicated to the memory of my father, Paul Joseph Sally.He was born in 1897 into an Irish family then living in Scotland. His familyimmigrated to the United States in 1907 and lived in Philadelphia.

He left school after the eighth grade and worked at Friends Hospital tohelp support his family. He joined the Army in 1917 and thereby obtained hiscitizenship. He fought in World War I and was among the first U.S. troopsthat landed in Europe. He was a highly intelligent man with unmatchedskills as a bricklayer, plasterer, and roofer. He knew the tools of his trade.

Paul J. Sally, Jr.May 2008

Table of Contents

Introduction ix

Acknowledgements xiii

Chapter 1. Sets, Functions, and Other Basic Ideas 1

§1. Sets and Elements 2

§2. Equality, Inclusion, and Notation 2

§3. The Algebra of Sets 4

§4. Cartesian Products, Counting, and Power Sets 8

§5. Some Sets of Numbers 10

§6. Equivalence Relations and the Construction of Q 15

§7. Functions 22

§8. Countability and Other Basic Ideas 30

§9. Axiom of Choice 38

§10. Independent Projects 41

Chapter 2. Linear Algebra 47

§1. Fundamentals of Linear Algebra 48

§2. Linear Transformations 54

§3. Linear Transformations and Matrices 56

§4. Determinants 59

§5. Geometric Linear Algebra 67

§6. Independent Projects 76

Chapter 3. The Construction of the Real and Complex Numbers 89

vii

viii Table of Contents

§1. The Least Upper Bound Property and the Real Numbers 90§2. Consequences of the Least Upper Bound Property 92§3. Rational Approximation 94§4. Intervals 97§5. The Construction of the Real Numbers 98§6. Convergence in R 102§7. Automorphisms of Fields 107§8. Construction of the Complex Numbers 108§9. Convergence in C 110§10. Independent Projects 115

Chapter 4. Metric and Euclidean Spaces 125§1. Introduction 125§2. Definition and Basic Properties of Metric Spaces 126§3. Topology of Metric Spaces 129§4. Limits and Continuous Functions 137§5. Compactness, Completeness and Connectedness 145§6. Independent Projects 155

Chapter 5. Complete Metric Spaces and the p-adic Completion of Q 167§1. The Contraction Mapping Theorem and Its Applications 168§2. The Baire Category Theorem and Its Applications 170§3. The Stone–Weierstrass Theorem 172§4. The p-adic Completion of Q 176§5. Challenge Problems 184

Index 189

Introduction

When structuring an undergraduate mathematics program, ordinarily thefaculty designs the initial set of courses to provide techniques that permita student to solve problems of a more or less computational nature. So,for example, students might begin with a one variable calculus course andproceed through multi-variable calculus, ordinary differential equations, andlinear algebra without ever encountering the fundamental ideas that underliethis mathematics. If the students are to learn to do mathematics well, theymust at some stage come to grips with the idea of proof in a serious way.

In this book, we attempt to provide enough background so that studentscan gain familiarity and facility with the mathematics required to pursue de-manding upper-level courses. The material is designed to provide the depthand rigor necessary for a serious study of advanced topics in mathematics,especially analysis.

There are several unusual features in this book. First, the exercises, ofwhich there are many, are spread throughout the body of the text. They donot occur at the ends of the chapters. Instead Chapters 1–4 close with specialprojects that allow the teachers and students to extend the material coveredin the text to a much wider range of topics. These projects are an integralpart of the book, and the results in them are often cited in later chapters.They can be used as a regular part of the class, a source of independentstudy for the students, or as an Inquiry Based Learning (IBL) experience inwhich the students study the material and present it to the class. At the endof Chapter 5, there is a collection of Challenge Problems that are intendedto test the students’ understanding of the material in all five chapters as wellas their mathematical creativity. Some of these problems are rather simplewhile others should challenge even the most able students.

ix

x Introduction

We now give an outline of the content of the individual chapters. Chap-ter 1 begins with set theory, counting principles, and equivalence relations.This is followed by an axiomatic approach to the integers and the presenta-tion of several basic facts about divisibility and number theory. The notionsof a commutative ring with 1 and a field are introduced. Modular systemsare given as examples of these structures. The ordered field of rationalnumbers is constructed as the field of quotients of the integers. Finally,cardinality, especially countability, is discussed. Several equivalent forms ofthe axiom of choice are stated and the equivalences proved.

Chapter 2 is about linear algebra. The first part of the chapter is devotedto abstract linear algebra up through linear transformations and determi-nants. In particular, the properties of determinants are attacked with bareknuckles. The final section of the chapter is devoted to geometric linearalgebra. This is a study of the algebra and geometry of Euclidean n-spacewith respect to the usual distance. It is a preparation for the study of metricspaces in Chapter 4 as well as for the geometric ideas that occur in advancedcalculus.

Chapter 3 begins with an axiomatic approach to the real numbers as anordered field in which the least upper bound property holds. Several fun-damental topics are addressed including some specific ideas about rationalapproximation of real numbers. Next, beginning with the rational numbersas an ordered field, the real numbers are constructed via the method ofequivalence classes of Cauchy sequences. After this construction, the stan-dard convergence theorems in the real numbers are proved. This includesthe one-dimensional versions of the Bolzano-Weierstrass theorem and theHeine-Borel theorem. The last sections involve the construction of the com-plex numbers and their arithmetic properties. We also study the topic ofconvergence in the complex numbers.

In Chapter 4, the stakes are raised a bit. There is a complete andthorough treatment of metric spaces and their topology. Such spaces asbounded real valued functions on a set with the sup norm, the infinite-dimensional �p spaces, and others are given careful treatment. The equiva-lence between compactness and sequential compactness is proved, and thestandard method of completing a metric space is presented. Here it is notedthat this process cannot be used to complete the rational numbers to thereal numbers since the completeness of the real numbers is fundamental tothe proof. At the end of the chapter, several topics such as convexity andconnectedness are analyzed.

Chapter 5 is a compendium of results that follow naturally from thetheory of complete metric spaces developed in Chapter 4. These results

Introduction xi

are essential in further developments in advanced mathematics. The Con-traction Mapping Theorem has a number of very useful applications, forexample, in the proof of the Inverse Function Theorem. We give an applica-tion to the solution of ordinary differential equations. The Baire Categorytheorem is most often used in functional analysis. We give an applicationto uniformly bounded families of continuous functions on a complete metricspace. The Stone–Weierstrass theorem concerns dense families of functionsin the algebra of continuous functions on a compact metric space. In par-ticular, this theorem implies the density of the polynomials in the algebraof continuous functions on closed bounded intervals in R. The final sectioncontains the most basic example of completing a metric space, that is, thep-adic completion of the rational numbers relative to a prime p. Along withbeing an example of the completion process, the p-adic completion yields afamily of locally compact fields that currently is prominent in research innumber theory, automorphic forms, mathematical physics, and other areas.

As pointed out above, each chapter ends with a set of special projectsthat are intended to broaden and deepen students’ understanding of ad-vanced mathematics. The first project in Chapter 1 is a series of exercisesin elementary number theory that serves as an introduction to the subjectand provides necessary material for the construction of the p-adic numbersin Chapter 5. Next, we introduce the idea of completely independent axiomsystems, so that students working through this project might have some ideaof the role of axioms in mathematics. Finally, we discuss ordered integraldomains. We ask the students to show that the integers, as an ordered in-tegral domain in which the Well Ordering Principle holds, are contained inevery ordered integral domain. This leads naturally to the conclusion thatevery ordered field contains the rational numbers.

The projects at the end of Chapter 2 provide a set of exercises for thestudent that form a primer on basic group theory, with special emphasis onthe general linear group and its subgroups.

The projects at the end of Chapter 3 present the students with an op-portunity to investigate the following topics: an alternate construction ofthe real numbers using Dedekind cuts; an introduction to the convergenceof infinite series; and a careful analysis of the decimal expansions of realnumbers. The material about the convergence of infinite series is used ex-tensively throughout the remaining chapters.

The projects in Chapter 4 provide an insight into advanced mathematics.They begin with an exploration for students of general point set topology,building on the theory of metric spaces covered in Chapter 4. Next, thestudents are asked to study a proof of the Fundamental Theorem of Algebrawhich establishes one of the basic facts in advanced mathematics.

xii Introduction

The first three chapters of this book are used in a one quarter transitioncourse at the University of Chicago. A substantial portion, but not all, ofthe material in the first three chapters can be covered in ten weeks. Theremaining material in the book is used in the first quarter of “Analysis inRn.” This course is intended as an advanced multivariable calculus coursefor sophomores. It covers geometric linear algebra from Chapter 2, someconvergence theorems in R and C in Chapter 3, and the theory of metricspaces in Chapter 4, with an introduction to Chapter 5 if time allows. Theremaining two quarters of Analysis in Rn cover differentiation theory andintegration theory in Rn along with the usual theorems in vector calculus.The entire book is more than sufficient for a two quarter or one semestercourse, and if the projects are covered completely there is more than enoughfor a three quarter or two semester course.

Acknowledgements

We acknowledge with gratitude the contributions to our book made by ourcolleagues, students, and friends over the span of twenty years or more.It is not possible to mention by name all those with whom we have hadconversations about this material, but we express appreciation to them here.

We thank Harvey Friedman, Bill Fulton, Dennis Hirschfeldt, David Leep,Raghavan Narasimhan, and Madhav Nori for their advice and assistance.

We thank John Conlon, Dan Gardner, Matt Gelvin, Grant Larsen, AlexMunk, Kevin Tucker and Shaffiq Welji for their careful, critical reading ofportions of the manuscript.

We are grateful to Sam Altschul, David Coley, Moon Duchin, Sam Isaac-son, Sean Johnson, Tom Koberda, Calvin Lin, Chris Malon, Emily Peters,Sam Raskin, and Ryan Reich for their incisive ideas on the material in thisbook, as well as their perceptive reading of portions of it.

We are in great debt to Mitya Boyarchenko who contributed significantideas to the book and to John Boller and Loren Spice, both of whom alsocontributed ideas to the book and carefully read the final manuscript.

My ultimate debt is owed to those who worked with me to produce thismanuscript. The word colleague describes them appropriately. The wordamanuensis could be used as a formal title, but they are much more. We ar-gued, discussed, rewrote, reaffirmed and readjusted parts of the manuscripton many occasions. These friends are Chris Jeris, Nick Ramsey, Kaj Gartz,Nick Ramsey (again), and Nick Longo. Without them, etc.

Paul J. Sally, Jr.Chicago, Illinois

May 2008

xiii

Index

(n − 1)-sphere, 133

2-sphere, 133

B3, 133

Bn, 133

S2, 133

Sn−1, 133

T0, 158

T1, 158

T2, 158

T3, 158

T4, 158

Q, 17

Qp, 179

Zn, 21

lim sup, 119

BC, 143

m × n matrix, 57

n-th root of unity, 114

p-adic absolute value, 176

p-adic completion of Q, 177

p-adic expansion, 181

p-adic field, 177

absolute value, 102

for C, 109

absolutely convergent series, 118

accumulation point, 104, 112, 132, 156

affine subspace, 69

algebra, 58

commutative, 58

with identity, 58

algebraic closure, 162

algebraic extension, 162

algebraic numbers, 115

algebraically closed, 114

angle

between vectors, 71

Archimedean ordered field, 93

Archimedean property, 92

associative, 24

associativity

of composition of functions, 24

automorphism

of a field, 108

automorphism group

of a field, 108

axiom, 43

axiom system, 43

base, 157

basis

of a topological space, 157

vector space, 51

bijective, 24

bilinear form, 69

positive definite, 70

symmetric, 70

binary operations, 19

binomial coefficient, 15

Binomial Theorem, 15

Bolzano-Weierstrass, 139

Bolzano-Weierstrass Lemma, 103

Bolzano-Weierstrass Theorem, 104

boundary, 132

boundary point, 132

bounded, 90, 142

bounded above, 90

bounded below, 90

bounded sequence

in C, 111

in Q, 99

in R, 103

189

190 Index

Bourbaki, 125

canonical basis, 51Cantor set, 105cardinal number

of a finite set, 30cardinality

aleph null (ℵ0), 31Cartesian product

(n-fold), 9countable union, 35of groups, 83

of sets, 8Cauchy sequence

in C, 111in Q, 98in R, 103in a metric space, 137

Chevalley, 47choice function, 39closed ball

closed unit ballin C, 110in R3, 133in Rn, 133

in a vector space, 129in a metric space, 129

closed convex hull, 136closed set

in C, 111in R, 105in a metric space, 131

in a topological space, 155closure, 134coefficients, 57commutative ring, 20commutative ring with identity, 20compact, 161

compact set

in C, 112in R, 106in a metric space, 145

complement, 5complete

metric space, 138

complete independence of axioms, 43completion, 154complex conjugate, 109complex numbers, 108complex power series, 120composite, 17composition

of functions, 24congruence, 16congruence modulo n, 41connected, 151connected component, 152

consistent, 43constant function, 22continuous, 140, 156Contraction Mapping, 168

convergenceconvergent sequence

in C, 110in R, 102

in a metric space, 137in Q, 98of partial sums, 117

convex, 136convex hull, 136

countable, 33countable Cartesian product, 35countably infinite, 36cross product, 72

cyclic groupfinite, 77infinite, 77

decimal expansion, 122Dedekind cut, 115

degree, 28DeMorgan’s laws, 6, 7, 132dense, 93, 150diagonal matrices, 81diameter, 135

difference of sets, 4dihedral group, 83dimension

of a vector space, 52

disconnected, 151discrete topology, 155disjoint sets, 4distance

between sets, 147divisibility, 16divisor, 16domain

of a function, 23

dot product, 70

element, 2empty set, 2epimorphism, 79equivalence class, 17

equivalence relations, 15even permutation, 61eventually constant sequence, 138exchange lemma, 51

factorial, 15

field, 20finite complement topology, 155finite set, 30, 34first category, 170

Index 191

fixed point, 168function

equality, 24

fundamental counting principle, 8

general linear group, 77, 81generalized Cantor set, 170glb, 38, 92Gram-Schmidt Orthogonalization, 75

greatest integer function, 27greatest lower bound, 38, 91group, 61, 76

abelian, 77

commutative, 77infinite, 78

group of units, 178

Hamel basis, 54

harmonic series, 118Hausdorff, 158Heine-Borel Theorem, 106, 112homeomorphism, 143, 156

homomorphism, 79hyperplane, 69

identity, function, 22, 25ILC, 19image, 23

imaginary part, 109inclusion–exclusion principle, 9independence of axioms, 43

index set, 7indiscrete topology, 155inf, 92infimum, 38

infinite set, 30, 34injective, 24integral domain, 20interior, 135

internal law of composition, 19intersection, 4interval, 97inverse

function, 26invertible matrix, 59irrational, 92isolated point, 132

isometry, 143isomorphism

of groups, 79

kernel

of a homomorphism, 80of a linear transformation, 55

latticeof functions, 174

lattice point, 185least upper bound, 38, 90length

of a vector, 70

limit, 102, 110, 137line, 69linear combination, 49linear independence, 50

linear isomorphism, 55linear operator, 54linear transformation, 54linearly dependent, 50Lipschitz condition, 169

locally compact, 161lower bound, 38, 90lub, 38, 92

mathematical induction, 14matrix, 57

maximal element, 38maximal ideal, 178metric space, 126Michelle function, 25

model, 43monomorphism, 79monotonic sequence

decreasing, 103increasing, 103

neighborhood, 97

in a topological space, 155non-Archimedean triangle inequality, 176norm

of a vector, 70

normaltopological space, 158

normal subgroup, 81normalized vector, 73

nowhere dense, 170

one-to-one, 24one-to-one correspondence, 24onto, 24open ball

in C, 110

in a metric space, 129open cover, covering, 106, 145open map, 157open set

in C, 111in R, 105in a metric space, 130in a topological space, 155

orderof a group, 78of an element, 78

order axioms, 12

192 Index

order isomorphic, 91ordered field, 21ordered integral domain, 21ordered pair, 8

orthogonal, 73vectors, 71

orthogonal group, 81orthogonal transformation, 84

orthonormal basis, 73

pairwise disjoint, 4

parallelepiped, 69partial sum, 117partially ordered set, 38perfect, 171

permutation, 60perpendicular, 73

vectors, 71Picard’s Theorem, 169

pigeon hole principle, 23plane, 69point, 67pointwise limit, 139

polar form, 113polynomial function, 28, 173power set, 10preimage, 26

prime, 16primitive root of unity, 114principal branch of the argument, 113principal units, 182

product topology, 160projection, 71proper subset, 2

quantifiers, 3quotient topology, 160

range, 24rational numbers, 17real algebraic numbers, 115real numbers, 91

real part, 109reduced residue system, 42regular

topological space, 158

relation, 8relative topology, 159ring of integers in Qp, 178rules of arithmetic, 10

scalar, 48scalar multiplication, 48scalar product, 70

Schroder-Bernstein Theorem, 31second category, 170semidirect product, 83

separable, 150, 157separates points, 172sequence, 29sequentially compact, 107, 147

setcontainment, 2equality, 2

shell, 179sign

of a permutation, 61span, 51spanning set, 51special linear group, 79, 81special orthogonal group, 85

standard basis, 50, 51Stone-Weierstrass Theorem, 173subfield, 20subgroup, 79

proper, 79

subintegral domain, 20subring, 20subset, 2subspace, 53sup, 92

superset, 2supremum, 38, 92surjective, 24symmetric difference, 5symmetric group, 61

symmetric neighborhood, 97

tail, 181topological space, 155topology, 155

generated by C, 157totally bounded, 148

totally disconnected, 152totally ordered set, 38transcendental numbers, 115trigonometric polynomial, 176

uniform convergence, 139uniformly continuous, 144

union, 4countable union, 35

unit ballclosed unit ball

in a metric space, 129

unit circle, 112unit interval

is uncountable, 36unit vector, 73universal set, 3

upper bound, 38, 90upper triangular matrices, 81upper triangular unipotent matrices, 81usual metric on Rn, 126

Index 193

usual topology on Rn, 155

valuation map, 180vector space, 48

finite dimensional vector space, 52infinite dimensional vector space, 53

well-defined, 18well-ordered set, 38well-ordering principle

for Z, 14

Zahlen, 10

This book provides a transition from the formula-full aspects of the beginning study of college level mathematics to the rich and creative world of more advanced topics. It is designed to assist the student in mastering the techniques of analysis and proof that are required to do mathematics.

Along with the standard material such as linear algebra, construc-tion of the real numbers via Cauchy sequences, metric spaces and complete metric spaces, there are three projects at the end of each chapter that form an integral part of the text. These projects include a detailed discussion of topics such as group theory, convergence of infinite series, decimal expan-sions of real numbers, point set topology and topological groups. They are carefully designed to guide the student through the subject matter. Together with numerous exercises included in the book, these projects may be used as part of the regular classroom presentation, as self-study proj-ects for students, or for Inquiry Based Learning activities presented by the students.

For additional information and updates on this book, visit

www.ams.org/bookpages/mbk-55

AMS on the Web www.ams.org

MBK/55