Norm Derivatives and Characterizations of Inner Product Spaces

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N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G T A I P E I C H E N N A I

World Scientifc

Claudi AlsinaUniversitat Politcnica de Catalunya, Spain

Justyna SikorskaSilesian University, Poland

M Santos TomsUniversitat Politcnica de Catalunya, Spain

Inner Product Spacesof

Norm DerivativesCharacterizations

and


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ISBN-13 978-981-4287-26-5

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All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,

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Published by

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USA office 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office 57 Shelton Street, Covent Garden, London WC2H 9HE

Printed in Singapore.

NORM DERIVATIVES AND CHARACTERIZATIONS OF INNER PRODUCT SPACES


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Preface

The aim of this book is to provide a complete overview of characterizations

of normed linear spaces as inner product spaces based on norm derivatives

and generalizations of the most basic geometrical properties of triangles in

normed linear spaces. Since the monograph by Amir that has appeared in

1986, with only a few results involving norm derivatives, a lot of papers have

been published in this field, many of them by us and our collaborators. So

we have decided to collect all these results and present them in a systematicway. In doing this, we have found new results and improved proofs which

may be of interest for future researchers in this field.

To develop this area, it has been necessary to find new techniques for

solving functional equations and inequalities involving norm derivatives.

Consequently, in addition to the characterizations of Banach spaces which

are Hilbert spaces (and which have their own geometrical interest), we trust

that the reader will benefit from learning how to deal with these questions

requiring new functional tools.This book is divided into six chapters. Chapter 1 is introductory and

includes some historical notes as well as the main preliminaries used in

the different chapters. The bulk of this chapter concerns real normed linear

spaces, inner product spaces and the classical orthogonal relations of James

and Birkhoff and the Pythagorean relation. In presenting this, we also fix

the terminology and notational conventions which are used in the sequel.

Chapter 2 is devoted to the key concepts of the publication: norm

derivatives. These functionals extend inner products, so many geometri-cal properties of Hilbert spaces may be formulated in normed linear spaces

by means of the norm derivatives. We develop a complete description of

their main properties, paying special attention to orthogonality relations

associated to these norm derivatives and proving some interesting char-

v


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vi Norm Derivatives and Characterizations of Inner Product Spaces

acterizations on the derivability of the norm from inner products. New

orthogonality relations are introduced and studied in detail, comparing

these orthogonalities with the classical Pythagorean, Birkhoff and Jamesorthogonalities.

Chapters 3, 4 and 5 are devoted to studying heights, perpendicular

bisectors and bisectrices in triangles located in normed linear spaces, re-

spectively. In doing a detailed study of the basic geometrical properties of

these lines and their associated points (orthocenters, circumcenters and in-

centers), we show a distinguished collection of characterizations of normed

linear spaces as inner product spaces. Chapter 6 is devoted to areas of

triangles in normed linear spaces.The book concludes with an appendix in which we present a series of

open problems in these fields that may be of interest for further research.

Finally, we list a comprehensive bibliography about this topic and a general

index.

This publication is primarily intended to be a reference book for those

working on geometry in normed linear spaces, but it is also suitable for

use as a textbook for an advanced undergraduate or beginning graduate

course on norms and inner products and analytical techniques for solvingfunctional equations characterizing norms associated to inner products.

We are grateful to Prof. Roman Ger (Katowice, Poland) for his positive

remarks, and to Ms Rosa Navarro (Barcelona, Spain) for her efficient typing

of the various versions of our manuscript.


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Special Notations

| | absolute value of a real number

\ difference of two sets

A(, ) angle

b(, ) generalized bisectrix

BX closed unit ball in a normed space

Bx(r) closed ball of radious r centered at x

composition of functionsdim dimension

, inner product

f1 inverse function off

inf infimum

i.p.s. inner product space

max maximum

min minimum

PA metric projection on AP Pythagorean orthogonality

A area orthogonality

B Birkhoff orthogonality

J James orthogonality

, orthogonality

R real line

R+ positive half-line

norm

vii


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viii Norm Derivatives and Characterizations of Inner Product Spaces

+, direction one-sided derivative of the square of the norm

+, direction one-sided derivative of

+,

, respectively

sgn generalized sign function sgn(x) := x/x for x = 0, sgn(0) = 0 e Euclidean norm

+ taxi-cab norm

diamond norm

n mixed norm

X generic vector space

(X, , ) generic i.p.s.

xy line determined by x, y

(X, ) generic real normed space[x, y] closed segment determined by x, y

(x, y) open closed segment determined by x, y

x, y inner product ofx, y

x norm ofx

Sx(r) sphere of radius r centered at x

SX unit sphere centered at 0

Kx(r) open ball of radius r centered at x

w(, ) vectorial bisectrix segment

[x]B the Birkhoffs orthogonal set ofx

[x] the -orthogonal set ofx


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Contents

Preface v

Special Notations vii

1. Introduction 1

1.1 Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Normed linear spaces . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Strictly convex normed linear spaces . . . . . . . . . . . . . 7

1.4 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Orthogonalities in normed linear spaces . . . . . . . . . . . 11

2. Norm Derivatives 15

2.1 Norm derivatives: Definition and basic properties . . . . . . 15

2.2 Orthogonality relations based on norm derivatives . . . . . 26

2.3 -orthogonal transformations . . . . . . . . . . . . . . . . 30

2.4 On the equivalence of two norm derivatives . . . . . . . . . 35

2.5 Norm derivatives and projections in normed linear spaces . 38

2.6 Norm derivatives and Lagranges identity in normed linear

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7 On some extensions of the norm derivatives . . . . . . . . . 45

2.8 -orthogonal additivity . . . . . . . . . . . . . . . . . . . . . 51

3. Norm Derivatives and Heights 57

3.1 Definition and basic properties . . . . . . . . . . . . . . . . 57

3.2 Characterizations of inner product spaces involving geomet-

rical properties of a height in a triangle . . . . . . . . . . . 60

ix


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x Norm Derivatives and Characterizations of Inner Product Spaces

3.3 Height functions and classical orthogonalities . . . . . . . . 74

3.4 A new orthogonality relation . . . . . . . . . . . . . . . . . 81

3.5 Orthocenters . . . . . . . . . . . . . . . . . . . . . . . . . . 853.6 A characterization of inner product spaces involving an

isosceles trapezoid property . . . . . . . . . . . . . . . . . . 91

3.7 Functional equations of the height transform . . . . . . . . 94

4. Perpendicular Bisectors in Normed Spaces 103

4.1 Definitions and basic properties . . . . . . . . . . . . . . . . 103


4.3 Relations between perpendicular bisectors and classicalorthogonalities . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.4 On the radius of the circumscribed circumference of a

triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.5 Circumcenters in a triangle . . . . . . . . . . . . . . . . . . 117

4.6 Euler line in real normed space . . . . . . . . . . . . . . . . 124

4.7 Functional equation of the perpendicular bisector transform 125

5. Bisectrices in Real Normed Spaces 131

5.1 Bisectrices in real normed spaces . . . . . . . . . . . . . . . 131


5.3 Functional equation of the bisectrix transform . . . . . . . . 144

5.4 Generalized bisectrices in strictly convex real normed spaces 149

5.5 Incenters and generalized bisectrices . . . . . . . . . . . . . 156

6. Areas of Triangles in Normed Spaces 163

6.1 Definition of four areas of triangles . . . . . . . . . . . . . . 163

6.2 Classical properties of the areas and characterizations of

inner product spaces . . . . . . . . . . . . . . . . . . . . . . 164

6.3 Equalities between different area functions . . . . . . . . . . 169

6.4 The area orthogonality . . . . . . . . . . . . . . . . . . . . . 172

Appendix A Open Problems 177

Bibliography 179

Index 187


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Chapter 1

Introduction

1.1 Historical notes

Functional analysis arose from problems on mathematical physics and as-

tronomy where the classical analytical methods were inadequate. For ex-

ample, Jacob Bernoulli and Johann Bernoulli introduced the calculus of

variations in which the value of an integral is considered as a function

of the functions being integrated, so functions became variables. Indeed,the word functional was introduced by Hadamard in 1903, and deriva-

tives of functionals were introduced by Frechet in 1904 [Momma (1973);

Dieudonne (1981)].

A key step in this historical development is precisely the contribution

made in 1906 by Maurice Frechet in formulating the general idea of metric

spaces, extending the classical notion of the Euclidean spaces Rn, so dis-

tance measures could be associated to all kinds of abstract objects. This

opened up the theory of metric spaces and their future generalizations, ex-tending topological concepts, convergence criteria, etc. to sequence spaces

or functional structures. In 1907, Frechet himself, and Hilberts student,

Schmidt, studied sequence spaces in analogy with the theory of square

summable functions, and in 1910, Riesz founded operator theory.

Motivated by problems on integral equations related to the ideas of

Fourier series and new challenges in quantum mechanics, Hilbert used dis-

tances defined via inner products.

In 1920, Banach moved further from inner product spaces to normedlinear spaces, founding what we may call modern functional analysis. In-

deed, the name Banach spaces is due to Frechet and, independently,

Wiener also introduced this notion. Banachs research [Banach (1922);

Banach (1932)] generalized all previous works on integral equations by

1


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2 Norm Derivatives and Characterizations of Inner Product Spaces

Volterra, Fredholm and Hilbert, and made it possible to prove strong re-

sults, such as the Hahn-Banach or Banach-Steinhaus theorems.

Abstract Hilbert spaces were introduced by von Neumann in 1929 inan axiomatic way, and work on abstract normed linear spaces was done by

Wiener, Hahn and Helly. In all these cases, the underlying structure of

linear spaces followed the axiomatic approach made by Peano in 1888.

The theory of Hilbert and Banach spaces was subsequently generalized

to abstract topological sets and topological vector spaces by Weil, Kol-

mogorov and von Neumann.

During the 20th century, a lot of attention was given to the problem of

characterizing, by means of properties of the norms, when a Banach space isindeed a Hilbert space, i.e., when the norm derives from an inner product.

While early characterizations of Euclidean structures were considered

by Brunn in 1889 and Blaschke in 1916, the first and most popular charac-

terization (the parallelogram law) was given by Jordan and von Neumann

in 1935. In subsequent years, Kakutani, Birkhoff, Day and James proved

many characterizations involving, among others, orthogonal relations and

dual maps, and Day wrote a celebrated monograph on this subject [Day

(1973)]. The topic became very active, as shown in [Amir (1986)], where350 characterizations are presented, summarizing the main contributions

up to 1986, such as those by Phelps, Hirschfeld, Rudin-Smith, Garkavi,

Joly, Bentez, del Ro, Baronti, Senechalle, Oman, Kircev-Troyanski, etc.

A lot of work has been done in the field of functional equations [Aczel

(1966); Aczel and Dhombres (1989)] to solve equations in normed linear

spaces where the unknown is the norm, and in this way new characteri-

zations have been found. It is also necessary to mention the interest in

orthogonally additive mappings developed by Pinsker, Drewnovski, Orlicz,

Sunderesan, Gudder, Strawther, Ratz, Szabo, etc. (and where in further

studies, the second author also made many contributions) as well as solv-

ing functional equations in normed linear spaces. We will be making use of

results and techniques arising in functional equations theory through this

book.

In a normed linear space (X, ), the norm derivatives are given for

fixed x and y in X by the two expressions

lim0

x + y x

.

The question [Kothe (1969)] of when a boundary point of the unit ball has

a tangent hyperplane is connected with the differentiability of the norm


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Introduction 3

at this point [Mazur (1933)], so norm derivatives have been considered

in problems looking for smooth conditions (see [Kothe (1969)], 26), but

very few characterizations of i.p.s. given in terms of norm derivatives werereported in [Amir (1986)]. Note that instead of considering the above norm

derivatives, it is more convenient to introduce the functionals

(x, y) = lim0

x + y2 x2

2= x lim

0

x + y x

because when the norm comes from an inner product , , we obtain

(x, y) = x, y,i.e., functionals are perfect generalizations of inner products. Our chief

concern in this publication is precisely to see how by virtue of these func-

tionals one can state natural generalizations of geometric properties of

triangles, and how by introducing new functional techniques one can obtain

a very large collection of new characterizations of norms derived from inner

products. In doing this, we report the latest results in the field and also

find new advances.

1.2 Normed linear spaces

We begin with the description of the well-known class of real normed linear

spaces

Definition 1.2.1 A pair (X, ) is called a real normed linear space

provided that X is a vector space over the field of real numbersR

and thefunction from X into R satisfies the properties:

(i) x 0 for all x in X,

(ii) x = 0 if and only if x = 0,

(iii) x = ||x for all x in X and in R,

(iv) x + y x + y for all x and y in X.

The function is called a norm and the real number x is said to

be the norm of x. In the real line R the only norms are those of the formx = |x|, x R, where | | denotes the absolute value |x| := max(x, x),

x R.

In general, for all x, y in X we havex y x y x + y, (1.2.1)


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so introducing the mapping d from X X into R by

d(x, y) := x y,

for all x, y in X, we infer that d is a metric induced by the norm , so

(X, d) is a metric space and therefore a topological space. With respect

to the metric topology, by virtue of (1.2.1), the norm is continuous

and the topology induced by the norm is compatible with the vector space

operations, i.e., R X (, x) x X and X X (x, y) x + y X

are continuous in both variables together.

The open ball Kx(r) of radius r centered at x consists of all points y in

X such that y x < r and can be obtained as the x-translation of theball K0(r) centered at the origin, i.e., Kx(r) = x + K0(r). Analogously, one

considers the closed ball

Bx(r) = {y X : y x r} = x + B0(r).

The sphere of radius r centered at x will be defined by

Sx(r) = {y X : x y = r},

and we denote the unit closed ball B0(1) by BX and the corresponding unit

sphere S0(1) by SX .

When all Cauchy sequences in (X, ) are convergent, i.e., the space

is complete, then the real normed space is said to be a Banach space.

Isometries in real normed spaces are characterized by Mazur and Ulam

(see, e.g., [Mazur and Ulam (1932); Benz (1994)]).

Theorem 1.2.1 Assume that (X, ), (Y, ) are real normed spaces

and let f be a surjective mapping from X onto Y which is an isometry, i.e.,

f(x) f(y) = x y,

for all x, y in X. Then the mapping T := f f(0) is linear.

Other interesting results on isometries on real normed spaces may be

found in [Benz (1992); Benz (1994)].

Let us recall the most characteristic examples of real normed linearspace.


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Introduction 5

(i) The space Rn

n 1 being fixed, and with the usual linear structure ofRn, one can

consider the following norms:

xe =

ni=1

x2i ,

x+ =ni=1

|xi|,

x = max{|x1|,..., |xn|},

xm = maxn1i=1

x2i1/2

, |xn|

for all x = (x1, x2,...,xn) in Rn. The norm e is the classical Euclidean

norm, + is the so-called taxi-cab norm, is the diamond norm and

m is a mixed norm.

(ii) The spaces c0, c and l

If l denotes the linear space of all bounded real sequences then itsnatural norm is defined by

x = sup{|xn| : n 1},

for any bounded real sequence x = (xn). One can restrict this norm to

the subspace c of all convergent real sequences and, in particular, to the

subspace c0 of all real sequences convergent to 0.

(iv) The space l1

In the linear space of real sequences (xn) such thatn=1

|xn| < one

considers the norm

x1 =n=1

|xn|

for all x = (xn

).


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(v) The spaces lp, 1 < p <

The classical inequality ab1 a + (1 )b for in (0,1) and

a, b 0, implies Holders inequality

n=1

|xnyn|

n=1

|xn|q

1/q n=1

|yn|p

1/p, (1.2.2)

whenever 1/p + 1/q = 1, 1 < p < , and real sequences (xn) and (yn) are

such that the right-hand side of (1.2.1) converges. From (1.2.1) one easily

derives Minkowskis inequalityn=1

|xn + yn|p

1/p

n=1

|xn|p

1/p

+

n=1

|yn|p

1/p

,

where 1 < p < +. Thus the space lp (1 < p < ) of infinite sequences

(xn) such thatn=1

|xn|p < admits the norm defined by

xp = n=1

|xn|p1/p

for all x = (xn).

(vi) The space C(K)

Given a compact space K, let C(K) be the vector space of all real-valued

continuous functions f defined on K. Then one considers the norm

f = sup{|f(x)| : x K}.

(vii) The Lp-spaces, p 1

A closed real interval [a, b] being fixed with a < b, for p 1, let Lp

denote the space of continuous real-valued functions defined on [a, b] and

such that

ba |f(t)|

p

dt < .

Then one defines the norm

fp =

ba

|f(t)|p dt

1/p.


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Introduction 7

1.3 Strictly convex normed linear spaces

An important class of real normed linear spaces is formed by the so-calledstrictly convex spaces. Precisely, we quote the following definition (see, e.g.,[Kothe (1969)]):

Definition 1.3.1 A real normed linear space (X, ) is said to be strictly

convexif in its closed unit ball BX every boundary point in BX is an extreme

point, i.e., any one of the following equivalent conditions holds:

(i) SX contains no line segments;

(ii) Every supporting hyperplane intersects SX in at most one point;

(iii) Distinct boundary points have distinct supporting planes;

(iv) If x = y = 1 and x = y then12

(x + y) < 1;

(v) Ifx + y = x + y and y = 0 then x = y for some 0;

(vi) If x, y in X are linearly independent then x + y < x + y.

The spaces (Rn, e), lp and Lp are strictly convex for 1 < p < and

for all n N, while (Rn, +), (Rn, ), (Rn, m), c0, l

1, l, L1, for

n > 1 are not.

1.4 Inner product spaces

Definition 1.4.1 A real vector space X is called an inner product space

(briefly i.p.s.) if there is a real-valued function , on X X that satisfies

the following four properties for all x,y,z in X and in R:

(i) x, x is nonnegative and x, x = 0 if and only if x = 0,

(ii) x, y + z = x, y + x, z,

(iii) x,y = x, y,

(iv) x, y = y, x.

An inner product , defined on X X induces the norm

x := x, x, x Xso all inner product spaces are normed linear spaces. When a norm is

induced by an inner product one says that the norm derives from an inner

product. In the case where (X, ) is a Banach space and derives

from an inner product , , then (X, , ) is called a Hilbert space.

One can show that any i.p.s. is strictly convex.


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In the Hilbert space (Rn, e) one considers the standard inner product

x, y =

nj=1

xjyj

for all vectors x = (x1,...,xn), y = (y1,...,yn) in Rn.

In the space L2 of continuous real-valued functions on [a, b] such thatba

|f(t)|2dt < ,

one may consider the inner product

f, g =

ba

f(t)g(t)dt.

In the space l2 one defines its inner product structure by means of the

expression

x, y =

n=1

xnyn

for all infinite sequences x = (xn) and y = (yn) in l2. A classical result

(see, e.g. [Reed and Simon (1972)]) states that, indeed, l2 is in some sense

the canonical example of a Hilbert space because any Hilbert space which

contains a countable dense set and is not finite-dimensional is isomorphic

to l2.

The classical result by Jordan and von Neumann [Jordan and von Neu-mann (1935)] states the following criterium for checking when the norm

derives from an inner product.

Theorem 1.4.1 Let (X, ) be a real normed linear space. Then

derives from an inner product if and only if the parallelogram law holds,

i.e.,

x + y

2

+ x y

2

= 2x

2

+ 2y

2

for all x, y in X.

The name of the law comes from its geometrical interpretation: the sum

of the squares of the lengths of the diagonals in a parallelogram equals the

sum of the squares of the lengths of the four sides.


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Introduction 9

There exist some modifications of Theorem 1.4.1, where the sign of

equality is replaced by one of the inequality signs, or where the condition

is satisfied for unit vectors only [Day (1947); Schoenberg (1952)].


derives from an inner product if and only if

x + y2 + x y2 2x2 + 2y2

for all x, y X, where stands either for or .


derives from an inner product if and only if

u + v2 + u v2 4

for all u, v SX , where stands for one of the signs =, or .

Theorem 1.4.4 [Bentez and del Rio (1984)] Let (X, ) be a real

normed linear space. Then derives from an inner product if and only

if for all u, v in SX there exist , = 0 such that

+ v2 + u v2 2(2 + 2),

where stands for =, or .

When the norm derives from an inner product , then the inner

product can be obtained from the norm by means of the polarization identity

x, y =1

4

(x + y2 x y2).

In our considerations, we will still require another classical characteri-

zation of inner product spaces

Theorem 1.4.5 Let(X, ) be a real normed linear space of dimension

greater than or equal to 2. The space X is an i.p.s. if and only if each

two-dimensional subspace of X is an i.p.s.

We now quote some additional geometrical notions which play a crucial

role in an i.p.s.

Definition 1.4.2 Two vectors, x and y, in an inner product space

(X, , ) are said to be orthogonal ifx, y = 0. A collection {xi} of vectors

in X is called an orthonormal set if xi, xi = 1 and xi, xj = 0 and for

all positive integers i, j and i = j.


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By virtue of orthonormal sets, one can formulate the following version

of the Pythagorean theorem.

Theorem 1.4.6 Let {x1,...,xn} be an orthonormal set in an i.p.s.

(X, , ). Then for all x in X and positive integers n, one has

x2 =ni=1

|x, xi|2 + x

ni=1

x, xixi2.

Therefore Bessels inequality follows

x2

ni=1

|x, xi|2

,

as well as the Cauchy-Schwarz inequality

|x, y| x y,

where equality holds if and only if x and y are linearly dependent.

Given a closed subspace H of a Hilbert space X, one considers the

orthogonal complement H

of H as the set of vectors of X which are or-thogonal to H. Then one obtains X = H H, and the fact that every

x in X may be uniquely written in the form x = u + v with u H and

v H constitutes the projection theorem.

IfS is an orthonormal set in a Hilbert space X and no other orthonormal

set contains Sas a proper subset then S is called an orthonormal basis. Such

bases always exist in Hilbert spaces, and ifS = {x| A} for some index

set A is one of them, then for all x in X we have

x =A

x, xx and x2 =

A

|x, x|2.

This last equality is called Parsevals relation and coefficients x, x are

often called the Fourier coefficients of x with respect to the basis.

Indeed, it is easy to show (Gram-Schmidt orthogonalization) that an

orthonormal set may be constructed from an arbitrary sequence of inde-

pendent vectors.

Finally, we recall the celebrated Riesz lemma which states that ifX is aHilbert space and X is its dual space of all bounded linear transformations

from X into R, then for any T in X there is a unique vector yT in X such

that T(x) = x, yT and, moreover,

yT = sup{|T(x)| : x = 1}.


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Introduction 11

1.5 Orthogonalities in normed linear spaces

In this section we recall the most classical orthogonality relations that havebeen studied in real normed spaces and whose properties have shown to

give interesting characterizations of i.p.s.

James orthogonality1

R.C. James (see [James (1945); James (1947a); James (1947b)]) intro-

duced in a real normed space (X, ) the following orthogonality relation:

x J y whenever x + y = x y,

using the idea that, in the plane, a practical way to examine the orthogo-

nality between two vectors x and y is to check whether the two diagonals

of the parallelogram determined by x and y are of equal length. When the

norm derives from an inner product , , then, James orthogonality

x J y reduces to the classical condition x, y = 0. Moreover the James

orthogonality J is symmetric (i.e., x J y if and only if y J x) and

partially homogeneous (i.e., if x J y, then ax J ay for all a in R).A large family of properties of J characterizes i.p.s. of dimension

greater than or equal to 2 (see [Amir (1986)]).

Theorem 1.5.1 Let (X, ) be a real normed linear space, dim X 2.

Each of the following conditions characterizes X as an i.p.s.:

(i) x J y implies x J ay for all x, y in X and for all a inR;

(ii) tu J v implies u J v for all u, v in SX , t = 0;

(iii) x J y implies x J y for all x, y in X and for some in (0, 1);(iv) x J z and y J z implies (x + y) J z for all x,y,z in X;

(v) The set {x X : x J z} is convex for all z in X.

Birkhoffs orthogonality2

In a real normed linear space (X, ), one says that a vector x is

orthogonal to y in the sense of Birkhoff [Birkhoff (1935)] if the following

relation holds:

x B y whenever x x + ty for all t in R.

1In the literature this orthogonality relation is also denoted by # instead ofJ (seee.g. [Amir (1986)]). It can also be called isosceles orthogonality.

2It is also called the Birkhoff-James orthogonality.


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This relation means, geometrically, that the line through x in the y-

direction supports the ball {z : z x} at x. Note that x B y yields

x B y for all , in R (i.e., B is full homogeneous). In an i.p.s. theBirkhoff orthogonality x B y is reduced to the usual condition x, y = 0.

Theorem 1.5.2 [Amir (1986)] Let (X, ) be a real normed linear

space, dim X 2. Each of the following conditions characterizes X as an

i.p.s.:

(i) x J y implies x B y for all x, y in X;

(ii) (u + v) B (u v) for all u, v in SX ;(iii)

x + x

yy

B

x xy

y

for all x, y in X, y = 0;

(iv) x B y implies x J y for all x, y in X;

(v) x B y implies x+y = F(x, y) for some F : R+R+ R+

and for all x, y in X;

(vi) If dim X 3, then x B z and y B z implies (x + y) B z for

all x,y,z in X;

(vii) If dim X 3, then x B y implies y B x for all x, y in X.

Pythagorean orthogonality

Taking into account the classical Pythagorean theorem, one can define

the orthogonal relation in a normed space (X, ):

x P y whenever x + y2 = x2 + y2.

As in the case of our previous orthogonalities, if the space considered isan i.p.s., the above definition x P y is reduced to the standard condition

x, y = 0. Moreover, P is symmetric, partially homogeneous and admits

diagonals (i.e., for all x, y = 0 there exists a unique t 0 with (x + ty) P(x ty)).

Theorem 1.5.3 [Amir (1986)] Let (X, ) be a real normed linear space,

dim X 2. Each of the following conditions characterizes X as an i.p.s.:

(i) x P y implies x P (y) for all x, y in X;

(ii) x P y implies x J y for all x, y in X;

(iii) x J y implies x P y for all x, y in X;

(iv) x P y implies x P y for all , inR and for all x, y in X;

(v) x P y implies x B y for all x, y in X;


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Introduction 13

(vi) x B y implies x P y for all x, y in X;

(vii) (u + v) P (u v) for all u, v in SX ;

(viii) x P z and y P z imply (x + y) P z for all x,y,z in X.


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Chapter 2

Norm Derivatives

We start this chapter by defining functions + and , which play a crucial

role in this book. Then we give several properties of these functions, which

are used for different characterizations of inner product spaces.

2.1 Norm derivatives: Definition and basic properties

Let (X, ) be a real normed linear space of at least dimension two. Weconsider the functions +,

: X X R defined as follows

(x, y) := limt0

x + ty2 x22t

, x, y X (2.1.1)

and call them norm derivatives.

In order to show that + and are well-defined, we recall a simple

fact (cf., e.g., [Kuczma (1985)]).

Lemma 2.1.1 Let I R be an open interval and let f : I R beconvex. Then for every x I there exist the right and left side derivatives.Moreover,

f(x) f+(x), x I.

Now we are able to show

Proposition 2.1.1 Functions given by (2.1.1) are well-defined.

Proof. Fix x, y X and define f : R R by

f(t) := x + ty2, t R.

15


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For arbitrary s, t R and [0, 1], by the convexity of functionx R x2 R we have

x + (t + (1 )s)y2 = (x + ty) + (1 )(x + sy)2 (x + ty + (1 )x + sy)2 x + ty2 + (1 )(x + sy)2= x + ty2 + (1 )x + sy2,

so f is convex.

Observe that

f

(0) = limt0 x + ty

2

x

2

t ,so

+(x, y) =1

2f+(0),

(x, y) =

1

2f(0) (2.1.2)

and on account of Lemma 2.1.1, for every x, y X limits exist.

Proposition 2.1.2 If (X,

,

) is a real inner product space, then both

+ and coincide with , .Proof. Take x, y X. Then

(x, y) = limt0

x + ty2 x22t

= limt0

x + ty,x + ty x, x2t

= limt0

2tx, y + t2y, y2t

= x, y.

Lemma 2.1.2 Function h : R \ {0} R defined for each fixed x, y Xby

h(t) :=x + ty x

t, t R \ {0}

is bounded and increasing on each of the intervals (, 0) and (0, +).Proof. Fix x, y

X. Since

x = x + ty ty x + ty + |t|y and x + ty x + |t|y,we have

y x + ty x|t| y,


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Norm Derivatives 17

whence

|h(t)

| y

, tR

\{0}

. (2.1.3)

Take t1, t2 R such that 0 < t1 t2. Thent2x + t1y = t2x + t1t2y = t1(x + t2y) + (t2 t1)x

t1x + t2y + (t2 t1)x,whence

x + t1y

x

t1 x + t2y

x

t2 ,which means that h is increasing in (0, +).

Now take t1, t2 R such that t1 t2 < 0. Thent1x + t2y = t1x + t1t2y = t2(x + t1y) + (t1 t2)x

t2x + t1y + (t2 t1)x,whence

x + t2y xt2

x + t1y xt1

,

which means again that h is increasing in (, 0). Example 2.1.1 Consider (R2, +). Then for (x1, x2), (y1, y2) R2 wehave

+((x1, x2), (y1, y2)) =

(x1 + x2)(y1 + y2) if x1x2 > 0,

(x1 x2)(y1 y2) if x1x2 < 0,x2y2 + |x2y1| if x1 = 0,x1y1 + |x1y2| if x2 = 0.

and

((x1, x2), (y1, y2)) =

(x1 + x2)(y1 + y2) if x1x2 > 0,

(x1 x2)(y1 y2) if x1x2 < 0,

x2y2 |x2y1| if x1 = 0,x1y1 |x1y2| if x2 = 0.These functionals can be written in the form (A. Monreal)

((x1, x2), (y1, y2)) = x1y1 + sgn(x1x2)(x1y2 + x2y1)

(1 |sgn(x1x2)|)(|x1y2| + |x2y1|) + x2y2.


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Still another form for [James (1945)] is:

((x1, x2), (y1, y2)) = (x1, x2)+ 2

i=1

xi=0

xi|xi|yi

xi=0

i{1,2}

|yi| .

Example 2.1.2 Consider (R2, ). Then for (x1, x2), (y1, y2) R2 wehave

+((x1, x2), (y1, y2)) =

x1y1, if |x1| > |x2|,x2y2, if

|x1

| 0,x1 min{y1, y2} if x1 = x2 < 0

and

((x1, x2), (y1, y2)) =

x1y1, if |x1| > |x2|,x2y2, if

|x1

| 0,x1 max{y1, y2} if x1 = x2 < 0.

Example 2.1.3 Consider the Banach space (lp, ) with p > 1 of allsequences x = (xn) such that

n=1|xn|p is convergent and with the norm

x =

n=1

|xn|p1/p

. Then ([James (1945)]) for all x = 0 we have

+(x, y) = (x, y) =

n=1

|xn|p2xnynxp2 .

This shows that all spaces lp with p > 1 are examples of smooth spaces,

which only in the case p = 2 are inner product spaces.

Example 2.1.4 Let c0 be the space of all sequences convergent to 0 withsupremum norm. Then for x = (xn), y = (yn) we have (see [Dragomir and

Koliha (2000)])

+(x, y) = sup|xn|=x

xnyn and (x, y) = inf

|xn|=xxnyn.


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Norm Derivatives 19

Example 2.1.5 Let C([0, 1]) be the space of all continuous real-valued

functions equipped with the norm x = sup{|x(u)| : u [0, 1]}. Then forfunctions x, y in C([0, 1]) we have (see [Dragomir and Koliha (1999)]):

+(x, y) = sup{x(u)y(u) : |x(u)| = x},(x, y) = inf{x(u)y(u) : |x(u)| = x}.

The next theorem describes several properties of + and (cf. [Amir

(1986); Lorch (1948); Tapia (1973)]).

Theorem 2.1.1 Let (X,

) be a real normed linear space at least two-

dimensional and let +, : X X R be given by (2.1.1). Then

(i) (0, y) = (x, 0) = 0 for all x, y X;

(ii) (x, x) = x2 for all x X;(iii) (x,y) =

(x,y) =

(x, y) for all x, y X and 0;

(iv) (x,y) = (x,y) =

(x, y) for all x, y X and 0;

(v) (x,x + y) = x2 + (x, y) for all x, y X and R;(vi)

|(x, y)

| x

y

for all x, y

X;

(vii) If X is strictly convex, then |(x, y)| < x y for all linearly inde-pendent x, y X;

(viii) (x, y) +(x, y) for all x, y X;(ix) +(x, y) =

+(y, x) for all x, y X or (x, y) = (y, x) for all

x, y X if and only if the norm in X comes from an inner product;(x) +(u, v) =

+(v, u) for all u, v SX or (u, v) = (v, u) for all

u, v SX if and only if the norm in X comes from an inner product.

Proof. (i) and (ii) follow directly from the definition of .

(iii) Take x, y X and 0. If = 0 then (0, y) = (x, 0) = 0.Assume that > 0. Then

(x,y) = limt0

x + ty2 x22t

= limt0

x + ty2 x2

2 t

= lims0

x + sy2 x2

2s= (x, y),

where s := t . Analogously,

(x,y)= limt0

x+ty2x22t

= lims0

x+sy2x2

2s= (x, y),

where s := t.


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(iv) The proof follows the same lines as in (iii) with the same substitu-

tions. Now t 0 if and only if s 0.(v) Take x, y X and R. If t is small enough then 1 + t > 0 and

(x,x + y) = limt0

x + t(x + y)2 x22t

= limt0

(1 + t)x + ty2 x22t

= limt0

(1 + t)2 x +t

1+ty2

x2 + (1 + t)2x2 x2

2t

= limt0

(1 + t)

x + t1+ty2 x2

2 t1+t+ lim

t0

(2t + 2t2)x22t

= lims0

11s

x + sy2 x22s

+ x2 = (x, y) + x2,

where s := t

1+t

, whence 1 + t = 1

1s

, and t

0 if and only ifs

0.

(vi) Take x, y X. From (2.1.3) we have limt0+

x + ty + x2

x + ty xt

= x limt0+

h(t)

x y,so

+(x, y)

xy, x, y X.

(vii) Observe first that in a strictly convex space for all linearly inde-

pendent vectors x and y, we have

x + y < x + y and x y < x + y. (2.1.4)From (v) and (iv) we have the following equalities:

+(x, x + y) = x2 + +(x, y) (2.1.5)

and

+(x, y x) = (x, x y) = x2 + +(x, y). (2.1.6)On account of (vi), equality (2.1.5) leads to

x2 + +(x, y) x x + y,


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Norm Derivatives 21

which from the linear independency of x and y and from (2.1.4) gives

+(x, y) x(x + y x) < x y.

Similarly, from (2.1.6) and (vi) we have

x2 + +(x, y) x x y.

Thus, the linear independency of x and y and (2.1.4) yield

+(x, y) x(x x y) > x y.These inequalities allow us to write

(x, y) = +(x, y) < x yfor all linearly independent vectors x and y, which completes the proof of

this property.

(viii) Follows from (2.1.2) and Lemma 2.1.1.

(ix) By means of Proposition 2.1.2, it is enough to show only one im-

plication. Assume that +(x, y) = +(y, x) for all x, y X, and observe

that this condition implies (x, y) = (y, x) for all x, y X, and con-

versely. Indeed, if, for example, +(x, y) = +(y, x) for all x, y X, then

on account of (iv) we obtain

(x, y) = +(x, y) = +(y, x) = (y, x).

Let P be any two-dimensional subspace of X. Define a mapping , :P P R by the formula

x, y := +(x, y) +

(x, y)

2, x, y P.

We will show that , is an inner product in P. It is easy to see that thefunction is symmetric, homogeneous and nonnegative. The only condition

which needs some explanation is the additivity of , in each variable.However, by the symmetry, it is enough to show the additivity with respect

to the second variable. Take x,y,z P. We consider two cases. Assumefirst that x and y are linearly dependent, so y = x for some R. From


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the earlier considerations and properties of , we get

x, y + z = x,x + z =+(x,x + z) +

(x,x + z)

2

= x2 + +(x, z) +

(x, z)

2= x,x + x, z = x, y + x, z.

Now let x and y be linearly independent, so z = x + y for some , R.Then

x, y + z

=

x,x + (1 + )y

=

+(x,x + (1 + )y) + (x,x + (1 + )y)

2

=x2 + (1 + )+(x, y) + x2 + (1 + )(x, y)

2

=+(x, y) +

(x, y)

2

+

x

2 + +(x,y) +

x

2 + (x,y)

2

= x, y + +(x,x + y) +

(x,x + y)

2= x, y + x, z.

So, we have proved that , defined in P is an inner product. By the freechoice of P, on account of Theorem 1.4.5, the norm in X comes from an

inner product.

(x) Assume that +(u, v) = +(v, u) for all u, v SX . Take arbitrary

x, y X\ {0} and let u := xx and v := yy . Of course, u, v SX , and wehave

+

x

x ,y

y

= +

y

y ,x

x

.

From (iii) we obtain

+(x, y) =

+(y, x).

Since +(x, 0) = +(0, x), the above equality holds for all x, y X, which

together with (ix) shows that the norm in X comes from an inner product.

Other properties of that we will use in the sequel are the following.


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Norm Derivatives 23

Proposition 2.1.3 The function+ is continuous and subadditive in the

second variable, and the function is continuous and superadditive in the

second variable.

Proof. Fix x0 X. From the convexity of function 2, for all y1, y2 Xand t R, we have2x0 + t(y1 + y2)2

2

12x0 + ty12 + 1

2x0 + ty22,

whence

2

x0 + t2 (y1 + y2)2 x02

x0 + ty12 x02

+(x0 + ty22 x02), (2.1.7)

and for t > 0 we get

x0 + t2 (y1 + y2)2 x022

t

2

x0 + ty12 x02

2t+

x0 + ty22 x022t

,

from which we obtain the subadditivity of + with respect to the second

variable

+(x0, y1 + y2) +(x0, y1) + +(x0, y2). (2.1.8)

In order to prove its continuity, we combine (2.1.8) with condition (vi) of

Theorem 2.1.1. Take h X. Then we have

+(x0, y + h) +(x0, y) +(x0, h) x0h.From the other side

+(x0, y + h h) +(x0, y + h) +(x0, h)= (x0, h) x0h.

This gives

+(x0, y + h) +(x0, y) x0h,which proves the continuity of + in the second variable.

The superadditivity in the second variable of function is obtained

from (2.1.7), after dividing it by 2t for t < 0 and taking the limit while

t 0.


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The continuity in the second variable of is obtained immediately

from the condition

(x0, y) = +(x0, y).

Proposition 2.1.4 [Amir (1986)] Let (X, ) be a real normed linearspace. Then there exists a set F X of Lebesgue measure zero such thatfor all x in X\F and y in X we have +(x, y) = (x, y), and X\F isdense in X.

Definition 2.1.1 [Kothe (1969); Precupanu (1978)] Let (X, ) be areal normed linear space. We say that X is smooth if the norm has theGateaux (or weak) derivative in X, i.e.,

G(x, y) := limt0

x + ty xt

exists for each x, y in X. If G(x, y) exists for an x in X and for each y inX, we say that X is smooth at x or that x is a point of smoothness.

Remark 2.1.1. Because is a convex function, the existence of theabove limit implies that G(x, ) is a linear function for each fixed x in X.Moreover, we may state Definition 2.1.1 in an equivalent form, namely:

X is smooth if and only if + = .

Proposition 2.1.5 In a real normed linear space (X, ) the followinginequality

|+(z, y + x) +(z, y)| ||xz (2.1.9)holds for all x,y,z in X and inR.


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Norm Derivatives 25

Proof. By the definition of the functional +:

|+(z, y + x) +(z, y)| = lim0+

z + (y + x)2

z2

2

lim0+

z + y2 z22

= lim0+

z + (y + x)2 z + y22

= lim0+

z + (y + x) + z + y

2z + (y + x) z + y

= z lim

0+

|z + y + x z + y|

z lim0+

||||x

= ||xz,

so (2.1.8) follows.

From (11) in [Precupanu (1978)] we can derive the next property of +.

Proposition 2.1.6 Let u, v SX . Then

lim0+

+(u + v,v) = +(u, v).

Remark 2.1.2. Employing the same argument as that used by Precupanu,

one can show that with the same assumptions

lim0

+(u + v,v) = +(u, v),

so, in fact we have

lim0 +(u + v,v) = +(u, v). (2.1.10)

Corollary 2.1.1 Let u, v SX . Then

lim0

(u + v,v) = (u, v). (2.1.11)


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Proof. By the properties of , we have

lim0

(u + v,v) = lim0

(

+(u + v,

v))

= lim0

+(u + ()(v), v)= +(u, v) = (u, v).

As a consequence of the above results and by properties of , one has

as even more general result.

Corollary 2.1.2 Let (X, ) be a real normed linear space withdim X 2. Functions R t (x + ty,y) R are continuous at zerofor every fixedx, y in X.

Finally, the functions characterize the Birkhoff orthogonality in the

following sense.

Proposition 2.1.7 [Amir (1986)] Let (X,

) be a real normed linear

space. Then for all x, y in X and in R, the condition x B y x issatisfied if and only if (x, y) x2 +(x, y).

2.2 Orthogonality relations based on norm derivatives

In this section we introduce two orthogonality relations based on the norm

derivatives and their connections with standard orthogonalities in normed

linear spaces.We start with the generalization of the usual notion of the orthogonality

in i.p.s. given by means of an inner product by defining in a normed space

the -orthogonal relation byx y if and only if +(x, y) = 0.

Proposition 2.2.1 Let (X,

) be a real normed linear space. The

relation satisfies the following conditions:(i) For all x in X, 0 x, x 0;

(ii) For all x in X, x x if and only if x = 0;(iii) For all x, y in X and for all nonnegative a, if x y then ax y and

x ay;


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Norm Derivatives 27

(iv) For all nonzero x, y in X, if x y then x and y are linearly inde-pendent;

(v) For all x in X and for all t > 0 there exists a y in X such thaty = tand x y;

(vi) If the norm derives from an inner product, then the relation isequivalent to the usual orthogonality in i.p.s.

Proof. We will prove (v), the rest follows easily from the definition of

. Fixed x in X and t > 0, define f from {z X : z = t} into R byf(z) := +(x, z). Then f is a continuous function with the property

f

t xx

= f

t xx

and therefore it is immediate to show that there exists a y in X such that

y = t and f(y) = 0, i.e., x y. Proposition 2.2.2 Let (X, ) be a real normed linear space. Thefollowing conditions are satisfied:

(i) For all x, y in X and inR

, x y x if and only ifx2 = +(x, y);

(ii) For all x, y in X, if x y, then x B y;(iii) If for all x, y in X, x y implies x P y, then X is an i.p.s.;(iv) If for all x, y in X, x y implies x J y, then X is an i.p.s.;(v) The condition x B y implies that x y holds for all x, y in X if

and only if X is smooth;

(vi) If for all x, y in X, x P y implies x y, then X is an i.p.s.;(vii) If for all x, y in X, x J y implies x y, then X is an i.p.s.;

(viii) If for all u, v in SX , u + v u v, then X is an i.p.s.Proof. We will prove (iii) and (v). For (iii), consider w, v SX such that+(w, v) =

(w, v) = 0.

Then w w v+(w,v) , and by hypothesis w P w v

+(w,v) , i.e.,

1 +w v+(w, v)

2

=2w v+(w, v)

2

. (2.2.1)

Moreover, w w + v+(w,v) and w P w + v

+(w,v) , i.e.,

1 +

w + v+(w, v)2

=1

+(w, v)2

. (2.2.2)


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By (2.2.1) and (2.2.2) we have 2w+(w, v) v = 1 for all w, v SX suchthat +(w, v) =

(w, v) (when

+(w, v) =

(w, v) = 0 the last equation

is obvious) and by (5.12) in [Amir (1986)], X is an i.p.s.In order to prove (v), assume first that X is smooth, so +(x, y) =

(x, y) for all x, y in X, and let x B y. Then by Proposition 2.1.7,(x, y) 0 +(x, y) and, consequently, (x, y) = +(x, y) = 0 ,whence x y.

For the converse, fix x, y in X. There exists a t in R such that

x B tx + y. Then +(x,tx + y) = 0, whence tx2 + +(x, y) = 0. Butwe also have

x

B tx + y, whence

t

x

2

(x, y) = 0. Consequently,

+(x, y) = (x, y) for all x, y in X.

Let us now define the -orthogonal relation by the conditionx y if and only if +(x, y) + (x, y) = 0.

Then we have analogous results to Propositions 2.2.1 and 2.2.2.

Proposition 2.2.3 Let (X, ) be a real normed linear space. Therelation

satisfies the following conditions:(i) For all x in X, 0 x, x 0;

(ii) For all x in X, x x if and only if x = 0;(iii) For all x, y in X and for all a, b inR, if x y then ax by;(iv) For all nonzero x, y in X, if x y, then x and y are linearly inde-

pendent;

(v) For all x in X and for all t > 0 there exists y in X such that y = tand x

y;

(vi) If the norm derives from an inner product, then the relation isequivalent to the usual orthogonality in i.p.s.

Proposition 2.2.4 Let (X, ) be a real normed linear space. Then:(i) For all x, y in X and inR, x y x if and only if

2x2 = +(x, y) + (x, y);

(ii) For all x, y in X and inR, if x

y x, then(x, y) x2 +(x, y);

(iii) For all x, y in X, if x y, then x B y;(iv) If for all x, y in X, x y implies x P y, then X is an i.p.s.;(v) If for all x, y in X, x y implies x J y, then X is an i.p.s.;


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Norm Derivatives 29

(vi) The condition x B y implies that x y holds for all x, y in X if andonly if X is smooth;

(vii) If for all x, y in X, x P y implies x

y, then X is an i.p.s.;(viii) If for all x, y in X, x J y implies x y, then X is an i.p.s.;

(ix) If for all u, v in SX , u + v u v, then X is an i.p.s.Proof. We will prove (iv) and (vi). For (iv), let w, v in SX be such that

+(w, v) = (w, v) = 0 (i.e., X is smooth at w). Then it is easy to check

that

w

w

v

+(w, v)

and w

w +

v

+(w, v)

so, by the assumption,

w P w v+(w, v)

and w P w + v+(w, v)

and

1 + w v

+(w, v)2

= 2w v

+(w, v)2

,

1 +

w + v+(w, v)2

=1

+(w, v)2

.

From the above two equalities, we deduce 2+(w, v)w v2 = 1 whenever+(w, v) = 0 (if +(w, v) = 0, last equality is trivial) and by (5.12) in[Amir (1986)], X is an i.p.s.

In order to prove (vi), take x, y in X. From the obvious inequality

(x, y) (x, y) + (1 )+(x, y)

x2 x2 +(x, y)

for every [0, 1], and by Proposition 2.1.7 we getx B tx + y

with t = (x,y)+(1)

+(x,y)

x2 . From the assumption we have

x tx + y,i.e.,

0 = +(x,tx + y) + (x,tx + y) = 2tx2 + +(x, y) + (x, y),


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whence

(1 2)(

(x, y)

+(x, y)) = 0

for each [0, 1], and consequently, +(x, y) = (x, y). The converseimplication is verified in the same way as the analogous condition for .

2.3

-orthogonal transformations

In the context of inner product spaces, the functional equation

T(x), T(y) = x, y corresponding to orthogonal transformations has re-ceived a lot of contributions in the literature. A natural generalization

of such equation in real normed linear spaces (X, ) is to look for thesolutions of

+(T(x), T(y)) = +(x, y) and

(T(x), T(y)) =

(x, y), (2.3.1)

for all x, y in X.

In the Euclidean plane, we have the following examples.

Example 2.3.1 Consider (R2, +), then T : R2 R2 verifies+(T(x), T(y)) =

+(x, y) if and only if T belongs to D4, the Euclidean

symmetry group of the square. This fact is referred to the result of[Schattschneider (1984)], where it was proved that the group of isometries

of the plane with respect to the taxi-cab metric is the semi-direct productof the group D4 and the group of all translations of the plane.

Example 2.3.2 Consider (R2, ), then T : R2 R2 verifies+(T(x), T(y)) =

+(x, y) if and only if T belongs to D4.

The next result is a technical lemma that we need for the main theorem.

Lemma 2.3.1 For any x in X, x = 0 there exists a nonzero vector y inX independent of x such that |+(x, y)| < x y.

Proof. Fix x X, x = 0. If for any vector z in X, z = 0, inde-pendent of x we would have |+(x, z)| = x z, then for any suchz, the vector x + z would also be independent of x and would satisfy


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Norm Derivatives 31

|+(x, x + z)| = x x + z, and consequently there would exist num-bers (z), (x + z) {1, 1} such that

(x + z)x x + z = +(x, x + z) = x2 + +(x, z) = x2 + (z)x z,

i.e.,

(x + z)x + z = x + (z)z.

Therefore, for all vectors z independent of x we would have

x + z2

= x2

+ z2

+ 2x z(z).Let z be a nonzero vector independent of x and x + z2 =

x2 + z2 (such a vector exists by the properties of Pythagorean or-thogonality (see [James (1945); Amir (1986)])). Then x z = 0 and weobtain a contradiction.

Next we prove the main result of this section [Alsina and Tomas (1991)].

Theorem 2.3.1 A continuous mappingT from a real normed linear space(X, ) of dimension two into itself verifies the equations:

+(T(x), T(y)) = g(x)+(x, y) and

(T(x), T(y)) = g(x)

(x, y)

(2.3.2)

for a function g from X into R satisfying the condition

g(x) = 0 implies x = 0

if and only if T is linear and there exists a positive constant k such thatg(x) = T(x)

2

x2= k for all x in X\{0}.

Proof. First we will show that T preserves the linear independence of any

couple of independent vectors x, y in X. In fact, if we had T(y) = T(x)

for some = 0, then T(y) = ||T(x) and by (2.3.2):

T(x)2 = +(T(x), T(x)) = +(T(x), T(y)) = g(x)+(x, y),

T(x)2

=

(T(x), T(x)) =

(T(x), T(y)) = g(x)

(x, y),T(x)2 = +(T(x), T(x)) = g(x)x2,

whence, since x = 0, we obtain that necessarily

=+(x, y)

x2 =(x, y)

x2 . (2.3.3)


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Moreover, for all z in X, z = 0 we have

+(z, y) =

1

g(z)

+(T(z), T(y)) =

1

g(z)

+(T(z), T(x))

=

g(z)sgn()(T(z), T(x)) =

sgn()(z, x) =

+(x, y)

x2 sgn()(z, x).

(2.3.4)

Let z := y +

xx

, y

xx

, which cannot be zero because of the in-

dependence of x and y. Then by (2.3.4) and the property (v) of + we

have:y +

x

x , y

x

x2

+ +

y +

x

x , y

x

x , +

x

x , y

x

x

= +

y +

x

x , y

x

x , y

= +

x

x , y

sgn()

y +

x

x , y

x

x ,x

x

.

Since, by (2.3.3), sgn() = sgn+ xx

, y, we deduce thaty + xx , y xx = 0, which gives a contradiction. Thus, T preservesthe linear independence.

Let F be the set of Lebesgue measure zero in X such that (x, y) =

+(x, y) for all x in X\F and for all y in X (see Proposition 2.1.4).Next we will show that T(x) = T(x) for all R and x in X\F,

x = 0. Take x in X\F. By Lemma 2.3.1 let us choose y independentof x and such that |

+(x, y)| < x y. Since T(x) and T(y) are alsoindependent, for every in R there exist , in R (depending on ) suchthat T(x) = T(x) + T(y).

Therefore, by (2.3.2) and the general properties of , we obtain the

equalities (in the case = 0 we assume sgn()(, ) = 0):

g(x)x2 = g(x)+(x,x) = +(T(x), T(x))= +(T(x), T(x) + T(y)) = T(x)2 + sgn()(T(x), T(y))

= g(x)x2

+ g(x)

sgn()(x, y)and, analogously,

g(y)sgn()(y, x) = g(y)+(y,x) =

+(T(y), T(x))

= +(T(y), T(x) + T(y))

= T(y)2 + sgn()(T(y), T(x))


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Norm Derivatives 33

= g(y)y2 + g(y)sgn()(y, x).Consequently, and satisfy the system of linear equations:

x2 + sgn()(x, y) = x2,sgn()(y, x) + y2 = sgn()(y, x).

The determinant of this system is

= x2y2 sgn()(y, x)sgn()(x, y) > 0,because using Lemma 2.3.1 and properties of ,

sgn()(y, x)sgn()(x, y)

sgn()(y, x) sgn()(x, y)=sgn()(y, x) +(x, y) < x2y2.

Then we have

=

x2y2 sgn()(y, x)sgn()(x, y)

and sgn() = sgn().

Consequently,

=x2

(sgn()(y, x) sgn()(y, x)) = 0 and = .

Thus we obtain T(x) = T(x) for all in R and x in X\F. Since Tis continuous and X\F is dense in X, we conclude that T(x) = T(x) forall x in X and in R.

Take e1 X\F and using Lemma 2.3.1 choose e2 in X, linearly inde-pendent ofx and such that |

(e1, e2)| < e1e2. If we consider x = v1e1and y = v2e2 for v1, v2 in R\{0}, then T(x), T(y) are independent vectorsand there exist a, b in R such that T(x + y) = aT(x) + bT(y). Therefore,

we obtain

g(x)(x2 + +(x, y)) = g(x)+(x, x + y) = +(T(x), T(x + y))= +(T(x), aT(x) + bT(y))

= aT(x)2 + bsgn(b)(T(x), T(y))

= ag(x)x2

+ bg(x)

sgn(b)(x, y)and

g(y)(y2 + +(y, x)) = g(y)+(y, x + y) = +(T(y), T(x + y))= +(T(y), aT(x) + bT(y))

= bT(y)2 + asgn(a)(T(y), T(x))


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= bg(y)y2 + ag(y)sgn(a)(y, x).Thus a, b satisfy the system of linear equations

ax2 + bsgn(b)(x, y) = x2 + +(x, y),asgn(a)(y, x) + by2 = y2 + +(y, x).

Since the associated determinant is > 0, we have

a =1

y2(x2 + +(x, y)) sgn(b)(x, y)(y2 + +(y, x))

0

and

b =1

x2(y2 + +(y, x)) sgn(a)(y, x)(x2 + +(x, y))

0

and consequently, a = b = 1 and T(x + y) = T(x) + T(y).

Finally, for all v in X there exist v1, v2 in R such that v = v1e1 + v2e2and

T(v) = T(v1e1 + v2e2) = T(v1e1) + T(v2e2) = v1T(e1) + v2T(e2)

and T is linear on X.Moreover, for all x, y in X, x = 0 and t in R, t = 0 such that x + ty = 0

since g(x) = T(x)2

x2, we have

g(x + ty) g(x)2t

=1

x2x + ty2

x2 T(x) + tT(y)2 T(x)2

2t

T(x)2 x + ty2 x2

2t

and we have

limt0

g(x + ty) g(x)2t

=1

x4x2(T(x), T(y)) T(x)2(x, y)

=

1

x4x2g(x) T(x)2 (x, y) = 0

and g is a constant function.

The converse implication has a trivial verification.

In the particular case where g 1 is the constant function, we have thefollowing result.

Corollary 2.3.1 A continuous mapping T from X into itself satisfies

(2.3.1) if and only if T is linear and T(x) = x for all x in X.


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Norm Derivatives 35

Corollary 2.3.2 A continuous mapping T from X into itself satisfies the

equations

+(T(x), T(y))T(x) T(y) =

+(x, y)x y and

(T(x), T(y))T(x) T(y) =

(x, y)x y (2.3.5)

for all nonzero x, y in X such that T(x) = 0, T(y) = 0, if and only if thereexists a function from X into R+ such thatT(x) = (x)(NL)(x), whereL : X X is linear, L(x) = x for all x in X and N is a functiondefined by N(x) := x

xfor all x in X, x = 0 and N(0) := 0.

Proof. Let L be the function L(x) =

T(x)x

T(x) , then L satisfy the hypothesisof Corollary 2.3.1: L is linear and L(x) = x for all x in X. ThenN(T(x) ) = (N L)(x) and this condition is equivalent to the equalityT(x) = (x)(N L)(x) for a function from X into R+.

From Theorem 2.3.1 and the last corollaries, we obtain the following

results.

Corollary 2.3.3 A continuous function T satisfies (2.3.1) and (2.3.5),

if and only if T is linear and T(x) = x for all x in X. (In this case(x) = x.)Corollary 2.3.4 A continuous function T satisfies (2.3.2) and (2.3.5),

if and only if there exist a constant k and a linear function L such that

L(x) = x for all x in X and T = kL. (In this case (x) = kx.)Corollary 2.3.5 A continuous function T satisfies (2.3.5) and it is ho-

mogeneous of degree 1, if and only if T(x) = (x)(N

L)(x), L is linear,

L(x) = x and (ax) = |a|(x) for all x in X and a inR.Corollary 2.3.6 A continuous function T satisfies (2.3.5) and it is in-

vertible, if and only if T(x) = (x)(N L)(x), L is linear, L(x) = xand (ax) = |a|(x) for all x in X and for all a inR, a = 1.

Some extensions of the previous results to semi-norms in R2 can be

found in [Alsina and Tomas (1991)].

2.4 On the equivalence of two norm derivatives

If X is a real linear space endowed with two norms 1 and 2, the twonorms are said to be equivalentwhenever their respective induced topologies

are the same. However, such a condition may be formulated in terms of a


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double inequality, namely, there exist two positive constants 0 < A Bsuch that

Ax2 x1 Bx2 (2.4.1)

holds for all x in X.

Since each norm i, i {1, 2}, has the associated functional (+)i,i {1, 2}, let us find out how condition (2.4.1) may be established in termsof the functionals (+)i. Let us denote these functionals by i, i {1, 2}.Then we have the following.

Theorem 2.4.1 In a real linear space, two norms 1 and 2 areequivalent if and only if there exists a positive constant such that

|1(x, y) 2(x, y)| min {x1y1, x2y2} (2.4.2)

for all x, y in X.

Proof. If (2.4.2) holds, the substitution y := x yields

x21 x21 x22 x21,x22 x21 x22 x22,

whence

x22 (1 + )x21 and x21 (1 + )x22,

i.e.,

11 +

x2 x1

1 + x2,

so (2.4.1) holds with A = 1/

1 + 1 + = B.Conversely, let us assume (2.4.1). Then we have

1(x, y) 2(x, y) x1y1 + x2y2 (B2 + 1)x2y2

and

1(x, y) 2(x, y) x1y1 x2y2 (B2 + 1)x2y2.

Therefore,

|1(x, y) 2(x, y)| (B2 + 1)x2y2.


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Norm Derivatives 37

In a similar way, one checks that

|2(x, y) 1(x, y)| 1 + 1A2 x1y1,so (2.4.2) follows by taking := max

1 +

1

A2, B2 + 1

> 1.

Remark 2.4.1. In the case where both functionals 1 and 2 are inner

products, by using the polarization identities i(x, y) = (x + y2i x y

2i )/4 for i

{1, 2

}, one obtains an alternative to (2.4.2) by means of the

inequality1(x, y) A2 + B22 2(x, y) B2 A24 (x22 + y22),

where A and B are the positive constants of equation (2.4.1).

Definition 2.4.1 Let X be a real linear space endowed with two norms

1 and

2, whose corresponding functionals (+)1 and (

+)2 are de-

noted by 1 and 2, respectively. Then 1 and 2 will be called equivalentfunctionals if there exist two constants A, B > 0, A B such that

A|1(x, y)| |2(x, y)| B|1(x, y)| (2.4.3)

for all x, y in X.

The next theorem clarifies the relation between (2.4.1) and (2.4.3).

Theorem 2.4.2 Under the above notations, 1 and 2 are equivalent if

and only if the norms 1 and 2 are equivalent and1(x, y)

x21=

2(x, y)

x22(2.4.4)

for all x, y in X, x = 0.

Proof. If (2.4.3) holds, then for all x, y in X, x = 0, and R, we haveA|1(x,x + y)| |2(x,x + y)| B|1(x,x + y)|, (2.4.5)

i.e.,

Ax21 + 1(x, y) x22 + 2(x, y) B x21 + 1(x, y) .


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The substitution := 1(x, y)/x21 yields at once

0 1(x, y)x21 x22 + 2(x, y) 0,i.e.,

1(x, y)

x21=

2(x, y)

x22,

so (2.4.4) holds. Moreover, the substitution y := x into (2.4.5) yields the

equivalence of the norms

Ax1 x2 Bx1.Conversely, assume that (2.4.1) is satisfied and consider the set

=

x21x22

: x X, x = 0

[A2, B2].

Since it is a non-empty bounded set, its infimum and supremum exist in

this interval

A2 inf sup B2.Consequently, we have for all x, y in X, x = 0

A2 |2(x, y)| inf |2(x, y)| x21

x22|2(x, y)| = |1(x, y)|

sup|2(x, y)| B2|2(x, y)|.

2.5 Norm derivatives and projections in normed linear

spaces

Equation (2.4.4), considered in the previous section, has a geometrical in-

terpretation. It is the generalized equality of the two projections

x, y1

x

21

=x, y2

x

22

because in a real inner product space (X, , ) with associated norm ,given two vectors x, y in X\{0}, the projection P{x}(y) of y on the lin{x}is the vector

P{x}(y) =x, yx2 x.


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Norm Derivatives 39

For the following results, we first give a definition of orthogonal sets.

Definition 2.5.1 Let (X,

) be a real normed linear space. We define

[x]B as being Birkhoffs orthogonal set of x, i.e.,

[x]B :=

y X : x B y = y X : (x, y) 0 +(x, y) .The next theorem clarifies completely the relation between condition

(2.4.4) and Birkhoffs orthogonal sets.

Theorem 2.5.1 Let X be a real normed linear space endowed with two

norms

i, i {

1, 2}

. Then, condition (2.4.4) holds if and only if [x]B

1=

[x]B2 for all x in X.

Proof. Assume that [x]B1 = [x]B2

for all x in X. Consider for x = 0 andany y in X, the associated real numbers i := (

+)i(x, y)/x2i , i {1, 2}.

Then, using Theorem 2.1.1 we have

()1(x, y) 1x21 = (+)1(x, y),

whence()1(x, y 1x) 0 (+)1(x, y 1x),

therefore y 1x [x]B1 , and consequently, y 1x [x]B2 . So, inparticular, 1x22 (+)2(x, y) and

(+)1(x, y)

x21 (

+)2(x, y)

x22.

Analogously one shows that y 2x [x]B2 = [x]B1 , and we obtain(+)2(x, y)

x22 (+)1(x, y)

x21 . Thus condition (2.4.4) holds.Conversely, if (2.4.4) is valid, then

()1(x, y)

x21=

(+)1(x, y)x21

=(+)2(x, y)

x22=

()2(x, y)

x22,

and consequently, if y [x]B1 , then we have

(

)2(x, y)x

21

x22 = (

)1(x, y) 0 (

+)1(x, y) = (

+)2(x, y)x

21

x22 ,i.e., y [x]B2 , so [x]B1 [x]B2 . The other inclusion follows at once.The proof is complete.

In what follows, we define the -orthogonal set.


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Definition 2.5.2 Let (X, ) be a real normed linear space. We call[x] the -orthogonal set of X if it is described by the expression

[x] =

y X : +(x, y) = 0

.

We will see that condition (2.4.4) is related to the orthogonal sets de-

scribed by two norms, as in the case of the Birkhoff orthogonality.

Theorem 2.5.2 Let X be a real normed linear space endowed with two

norms i, i {1, 2}. Then condition (2.4.4) holds if and only if [x]1 =[x]2 for all x in X.

Proof. It is obvious that (2.4.4) implies [x]1 = [x]2

. Conversely, if we

start with the assumption [x]1 = [x]2

for all x in X, let us consider

any x, y in X, x = 0 and the real number = (+)1(x, y)/x21. Then(+)1(x,x + y) = x21 + (+)1(x, y) = 0,

so x + y [x]1 , and it must belong to [x]2

, i.e.,

0 = (+)2(x,x + y) = x

22 + (

+)2(x, y)

= (+)1(x, y)x22x21

+ (+)2(x, y),

i.e., (2.4.4) follows.

Another interesting result combining (2.4.4), the -orthogonality andBirkhoff orthogonality is given in the following.

Theorem 2.5.3 Let X be a real normed linear space endowed with twonorms i, i {1, 2}. Assume that(+)1 = ()1. Then condition (2.4.4)holds if and only if for all x in X we have [x]1 = [x]

B2

.

Proof. If (2.4.4) holds then it is obvious that [x]1 [x]B2 . In the otherdirection, if y [x]B2 , then by Theorem 2.5.1 we have y [x]B1 , and byvirtue of condition (+)1 = (

)1 we obtain y [x]1 . Conversely, assume

that [x]1 = [x]B2

. Given x, y in X, x = 0, take := (+)2(x, y)/x22.Then

(+)2(x,x + y) = x22 + (+)2(x, y) = 0 ()2(x,x + y),i.e., x + y [x]B2 , so x + y [x]

1

and we have

0 = (+)1(x,x + y) = x21 + (+)1(x, y)


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Norm Derivatives 41

= (+)2(x, y)x121x222

+ (+)1(x, y),

and (2.4.4) holds.

2.6 Norm derivatives and Lagranges identity in normed

linear spaces

In three-dimensional inner product space (R3, , ) one has the cross prod-uct

satisfying, among others, the bi-additive conditions

x (y + z) = x y + x z and (x + y) z = x z + y z

and the well-known Lagrange identity

x y, z v = x, zy, v x, vy, z.

Then, as a natural generalization, in a real normed linear space (X, )

of dimension 3 we consider the following problem: determine functions Ffrom X X into X satisfying the following conditions for all x,y,z,v in X:

F(x, y + z) = F(x, y) + F(x, z), (2.6.1)

F(x + y, z) = F(x, z) + F(y, z), (2.6.2)

and Lagranges identity in normed linear spaces:

+(F(x, y), F(z, v)) = +(x, z)

+(y, v) +(x, v)+(y, z). (2.6.3)

Note, in particular, that (2.6.3) implies by taking z := x and v := y

that

F(x, y)2 = x2y2 +(x, y)+(y, x). (2.6.4)

Lemma 2.6.1 If F satisfies (2.6.1), (2.6.2) and (2.6.4), then

(i) F(x, x) = 0 for all x in X; (2.6.5)

(ii) F(y, x) = F(x, y) for all x, y in X; (2.6.6)(iii) F(x,ay + bz) = aF(x, y) + bF(x, z) for all real a, b and x,y,z in X.

(2.6.7)


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Proof. The substitution of y := x into (2.6.4) yields (i). Next, by (i),

F(x + y, x + y) = 0 and by (2.6.1) and (2.6.2), one gets (ii). Finally, by

(2.6.4) and the properties of +, F(x, ) is continuous at y = 0, and by

(2.6.1) condition (iii) follows.

Lemma 2.6.2 If F satisfies (2.6.1), (2.6.2) and (2.6.3), then

+(x, y) = (x, y) for all x, y in X. (2.6.8)

Proof. By (2.6.3) and Lemma 2.6.1, we have

0 = +(F(x,

y), F(y,

y)) = +(x, y)

+(

y,

y)

+(x,

y)+(

y, y)

= +(x, y)y2 ((x, y))((y, y)) = (+(x, y) (x, y))y2,whence for y = 0, +(x, y) = (x, y), and since this last equality is obviousfor y = 0, we can conclude (2.6.8).

Theorem 2.6.1 If (X, ) is a real normed linear space of dimension 3,and there exists a function F from X X into X satisfying (2.6.1), (2.6.2)and (2.6.3), then necessarily the norm is induced by an inner product.Proof. Assume that F from X X into X satisfies (2.6.1), (2.6.2) and(2.6.3). By the previous lemmas for all x, z in X and a, b in R, we have

F(z, x + bz)2 = F(z, x + az + bz)2,i.e.,

z2x + bz2 +(z, x + bz)+(x + bz,z)=

z

2

x + (a + b)z

2

+(z, x + (a + b)z)

+(x + (a + b)z, z). (2.6.9)

We can rewrite (2.6.9) in the form

z2x + (a + b)z2 x + bz2 = +(z, x + (a + b)z)+(x + (a + b)z, z) +(z, x + bz)+(x + bz,z). (2.6.10)

Let us fix x and z, two independent vectors in X, and let us introduce

the function f from R into R defined by

f(t) := +(x + tz,z) = (x + tz,z). (2.6.11)

Thus, by means of (2.6.10) and (2.6.11) we can write

z2 x + (a + b)z2 x + bz2=

(a + b)z2 + +(z, x)

f(a + b) bz2 + +(z, x) f(b)


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Norm Derivatives 43

=

bz2 + +(z, x)

[f(a + b) f(b)] + az2f(a + b). (2.6.12)Since the norm is continuous and

|f(a + b)

| x + (a + b)z

z

, taking

limits in (2.6.12) when a 0, we obtainbz2 + +(z, x)

lima0

(f(a + b) f(b)) = 0,

i.e., for any real b, b = b0 := +(z, x)/z2, we obtainlima0

f(b + a) = f(b). (2.6.13)

Note that by (2.6.12) and (2.6.11) at point b0 we have

lima0

f(b0 + a) = lima0

x + b0z + az2 x + b0z2a

= 2(x + b0z, z) = 2f(b0). (2.6.14)

We claim that f(b0) = 0. To verifies this, consider the following chain

of equalities for any real and for our fixed x, z:

2

F(x, z)

2 =

F(x,z)

2 =

F(x + b0z,z)

2

= F(x + b0z, x + b0z + z)2

= x + b0z2x + b0z + z2

+(x + b0z, x + b0z + z)+(x + b0z + z,x + b0z + z z)

=

x + b0z2

x + b0z + z

2

x + b0z2 + +(x + b0z,z) x + b0z + z2+ +(x + b0z + z, z)

= x + b0z2+(x + (b0 + )z, z) x + b0z + z2+(x + b0z,z)

+(x + b0z,z)

+(x + (b0 + )z, z). (2.6.15)

Since by Lemma 2.6.2 we have + = , division by < 0 in (2.6.15)

yields

F(x, z)2 = x + b0z2+(x + (b0 + )z, z)


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x + b0z + z2+(x + b0z, z)+ +(x + b0z, z)

+(x + (b0 + )z, z) (2.6.16)

and taking limits when 0, by using (2.6.14) we obtain

0 = x + b0z22f(b0) x + b0z2f(b0),

and since x and z are independent, f(b0) = 0. Therefore, by (2.6.16), for

any < 0 it is

F(x, z)2

= x + b0z2

f(b0 + ),

i.e., for any t < b0:

f(t) = f(b0 + (t b0)) = F(x, z)2

x + b0z2 (t b0),

so f is an affine function on (, b0]. Since f(b0) = 0, by (2.6.15) we alsohave for > 0

2F(x, z)2 = x + b0z2f(b0 + ),

i.e., for any t > b0:

f(t) = f(b0 + (t b0)) = F(x, z)2

x + b0z2 (t b0),

so f is an affine function on R vanishing at b0. Thus for all real t

+(x + tz,z) =F(x, z)2x +(z,x)z2 z2

t +

+(z, x)

z2

,

and for t = 0 we obtain:

+(x, z) =x2z2 +(x, z)+(z, x)

x

+(z,x)

z2 z2

+(z, x)

z2

,

i.e., +(x, z) = 0 if and only if +(z, x) = 0 and the symmetry of the

orthogonality relation +(x, z) = 0 yields that necessarily (in dimension 3)

the norm derives from an inner product (see (19.6) in [Amir (1986)]). This

completes the proof.


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Norm Derivatives 45

2.7 On some extensions of the norm derivatives

Our aim in this section is to consider orthogonal relations in situationswhere we use functionals satisfying weaker conditions than those quoted

above for +, and also to check in which cases the required properties force

the derivability of the norm from an inner product. Precisely, let (X, )be a real normed linear space, and let F be a function from X X into Rsuch that

(i) F(x, x) = 0 = F(0, 0) whenever x = 0;(ii) There exist > 0 and a function f : R+

R+ such that for all x, y in

X and > 0:

|F(x, x + y) F(x, x) f()F(x, y)| ;(iii) There exists > 0 such that for all x, y in X and > 0

|F(x,y) F(x,y)| .The above set of conditions (i), (ii) and (iii) is a weaker requirement

than that satisfying the corresponding properties of from Theorem 2.1.1.The following results will in fact show a kind of stability of generalization

of conditions (v) and (iii) for Theorem 2.1.1.

Remark 2.7.1. Conditions (i), (ii) and (iii) are independent even in di-

mension 1. To verify this, consider the following examples:

(a) The function F(x, y) = 0 for all x, y satisfies (ii) and (iii), but not (i).

(b) Given > 0, the function F : R R R defined by F(x, y) := x2

, ifx = y and F(x, y) :=min(xy,) if x = y, satisfies (i) and (iii), but doesnot satisfy (ii) because there is no function f such that

| min(x2 + xy,) x2 f()min(xy,)| ,since the term on the left-hand side tends to infinity when x tends to

infinity.

(c) The function F : R

R

R, F(x, y) = y satisfies (i), (ii) with f() = ,

for all , but (iii) fails.

We begin with a technical but crucial result.

Lemma 2.7.1 Let (X, ) be a real normed linear space withdim X 2. If a functional F : X X R satisfies (i), (ii), (iii) and


56/196


the condition

F(x,x)(F(x,x)

x2) = 0, (2.7.1)

for all x in X and > 0, then necessarily

F(x,y) = F(x,y) = F(x, y), (2.7.2)

and

F(x, x + y) = x2 + F(x, y), (2.7.3)

for all x, y in X and > 0.

Proof. Assume that F satisfies (i), (ii), (iii) as well as condition (2.7.1).

By (2.7.1) with = 1 and using (i), we deduce F(x, x) = x2 for all x inX. Therefore, by (ii) with x = y, for any > 0 we have

|F(x, (1 + )x) (1 + f())x2| and hence,

limx

F(x, (1 + )x)

x2 = 1 + f() (2.7.4)

for every x X, x = 0. It follows that the inequality F(x, (1 + )x) > 0 issatisfied for x large enough, thus by (2.7.1) we also have

limx

F(x, (1 + )x)

x2 = limx(1 + )x2

x2 = 1 + . (2.7.5)

From (2.7.4) and (2.7.5) we get f() = for every > 0.

Next, for b > 0 the substitution y := bz into (ii) implies

|bF(x, z) F(x,bz)| |bF(x, z) + x2 F(x, x + (b)z)|

+ |F(x, x + (bz)) x2 F(x,bz)| 2,

i.e., |bF(x, z) F(x,bz)| 2/, and letting tend to infinityF(x,bz) = bF(x, z), (2.7.6)

for all b > 0. Consequently, by (iii) we obtain

|F(bx,z) bF(x, z)| = |F(bx,z) F(x,bz)| .


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Norm Derivatives 47

Dividing by b and taking limit when b tends to infinity we get

limb

F(bx,z)

b = F(x, z),

i.e., for all a > 0

F(ax,z) = limb

F(bax, z)

b= lim

ba

F(bax, z)

ba= aF(x, z). (2.7.7)

Thus, (2.7.6) and (2.7.7) prove (2.7.2). Finally, taking x := nu and y := nv

in (ii):

|F(nu,nu + nv) n2||u2 F(nu,nv)| ,but using (2.7.6) and (2.7.7):

|F(u, u + v) u2 F(u, v)| /n2,so letting n tend to infinity, we obtain (2.7.3).

Theorem 2.7.1 Let (X, ) be a real normed linear space withdim X

2. A functional F : X

X

R satisfies (i), (ii), (iii) and

the Pythagorean identity

(iv) x2 =F(y, x)y2 y

2

+

x F(y, x)y2 y2

for all x, y in X, y = 0, if and only if X is an inner product space whoseinner product associated with the norm is precisely F.Proof. Assume that F satisfies (i), (ii), (iii) and (iv). Then

x2y4 = [F(y, x)]2y2 + y2x F(y, x)y2 ,and for x := y, > 0 we obtain

F(y,y)(F(y,y) y2) = 0,that is (2.7.1), so we can apply the previous lemma and induce the validity

of both (2.7.2) and (2.7.3). Bearing this in mind, and substituting y := u

and x := u + v, > 0 into (iv), we get

u + v2u4 = [F(u, u + v)]2u2 + u2(u + v) F(u, u + v)u2

= (u2 + F(u, v))2u2

+ u2(u + v) u2u F(u, v)u2,


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whence

u4(

u + v

2

u2) = 2[F(u, v)]2

u2 + 2F(u, v)

u4

+ 2u2v F(u, v)u2, (2.7.8)and dividing by 2 and taking limits when 0+, we obtain

+(u, v) = F(u, v). (2.7.9)

Finally, using (2.7.9), (2.7.8) with = 1 and (iv),

u

4(

u + v

2

u

2) = [+(u, v)]

2

u

2 + 2+(u, v)

u

4

+ u2v +(u, v)u2

= [+(u, v)]2u2 + 2+(u, v)u4

+v2u4 [+(u, v)]2u2

= 2+

(u, v) +

v2 u4,

i.e.,

u + v2 = u2 + v2 + 2+(u, v),so + is symmetric, and therefore it must be an inner product. The converse

is immediate.

Remark 2.7.2. If (i) or (ii) do not hold, we may still have (iii) and (iv)for a norm derivable from an inner product, but different from F. The

following examples show why this is possible.

Example 2.7.1 F 0 satisfies (ii), (iii) and (iv) but does not verify (i).

Example 2.7.2 In the Euclidean space (R2, ), if denotes the char-

acteristic function of the real set [0, 1] Q

, define F :R2

R2

R

bymeans of

F(x, y) = (|x y|/xy)(x y).Then it is easy to see that F satisfies (i), (iii) for any > 0 and (iv).

If F satisfied (ii) then x = y would yield f() = for all > 0, and


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Norm Derivatives 49

substituting x = (3n, 4n), y = (1, 0) one would arrive at an inequality,

where the right-hand side is and on the left-hand side the term may tend

to infinity when n grows to infinity.

Now we turn our attention to the James orthogonality.

Theorem 2.7.2 Let (X, ) be a real normed linear space withdim X 2. A functional F : X X R satisfies (i), (ii), (iii) andthe James identity

(v) x = x 2F(y, x)

y2 y for all x, y in X, y = 0,if and only if X is an i.p.s. whose inner product is F.

Proof. Assume that F satisfies the above mentioned conditions.

Using (v) with x = y, > 0, and bearing in mind (i), we obtain

F(y,y) = 0 or F(y,y) = y2,

i.e., (2.7.1) holds, and by Lemma 2.7.1 we have (2.7.2) and (2.7.3). Sub-stituting y := u and x := u + v, > 0, into (v) and using (2.7.3), we

get

u + v =u +

2

F(u, v)

u2 u v , (2.7.10)

whence

u + v2 u22

=

u + 2F(u,v)u2 u v2 u22

.

So, takin

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Norm Derivatives and Characterizations of Inner Product Spaces