Ebook norm derivatives and characterizations of inner product spaces part 1

Inner Product Spaces of Norm Derivatives Characterizations and This page intentionally left blankThis page intentionally left blank 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[.]

Inner Product Spacesof

Norm Derivatives

This page intentionally left blank

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 • TA I P E I • C H E N N A I

World Scientific

Claudi Alsina

Universitat Politècnica de Catalunya, Spain

Justyna Sikorska

Silesian University, Poland

M Santos Tomás

Universitat Politècnica de Catalunya, Spain

Inner Product Spaces of

Norm Derivatives

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

ISBN-10 981-4287-26-1

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Copyright © 2010 by World Scientific Publishing Co Pte Ltd.

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NORM DERIVATIVES AND CHARACTERIZATIONS OF INNER PRODUCT SPACES

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 systematic way 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

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 James orthogonalities

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 solving functional 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

Special Notations

A±(·, ·) angle

b±(·, ·) generalized bisectrix

⊥ρ, ⊥ρ ρ′

vii

viii Norm Derivatives and Characterizations of Inner Product Spaces

ρ′

+, ρ′

ρ′′

+, ρ′′

+, ρ′

−, respectively

(X, h·, ·i) generic i.p.s

1.1 Historical notes 1

1.2 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 ρ′ ±-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 Lagrange’s identity in normed linear spaces 41

2.7 On some extensions of the norm derivatives 45

2.8 ρ-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

x Norm Derivatives and Characterizations of Inner Product Spaces

3.6 A characterization of inner product spaces involving an

4.1 Definitions and basic properties 103 4.2 A new orthogonality relation 106 4.3 Relations between perpendicular bisectors and classical

orthogonalities 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

5.1 Bisectrices in real normed spaces 131 5.2 A new orthogonality relation 136 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

6.1 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

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 Fr´echet in 1904 [Momma (1973); Dieudonn´e (1981)]

A key step in this historical development is precisely the contribution made in 1906 by Maurice Fr´echet in formulating the general idea of metric

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, Fr´echet himself, and Hilbert’s 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 normed linear spaces, founding what we may call modern functional analysis In-deed, the name “Banach spaces” is due to Fr´echet and, independently, Wiener also introduced this notion Banach’s research [Banach (1922); Banach (1932)] generalized all previous works on integral equations by

1

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 in

an 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 is indeed 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)], where

350 characterizations are presented, summarizing the main contributions

up to 1986, such as those by Phelps, Hirschfeld, Rudin-Smith, Garkavi, Joly, Ben´ıtez, del R´ıo, Baronti, Senechalle, Oman, Kirˇcev-Troyanski, etc

A lot of work has been done in the field of functional equations [Acz´el (1966); Acz´el 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, R¨atz, Szab´o, 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 maksolv-ing use of results and techniques arising in functional equations theory through this book

In a normed linear space (X, k · k), the norm derivatives are given for fixed x and y in X by the two expressions

lim

λ→0±

kx + λyk − kxk

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

a tangent hyperplane is connected with the differentiability of the norm

derivatives, it is more convenient to introduce the functionals

ρ′±(x, y) = lim

λ→0±

λ→0±

kx + λyk − kxk λ because when the norm comes from an inner product h·, ·i, we obtain

ρ′±(x, y) = hx, yi, i.e., functionals ρ′

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

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

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

provided that X is a vector space over the field of real numbers R and the function k · k from X into R satisfies the properties:

(i) kxk ≥ 0 for all x in X,

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

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

(iv) kx + yk ≤ kxk + kyk for all x and y in X

The function k · k is called a norm and the real number kxk is said to

be the norm of x In the real line R the only norms are those of the form kxk = |x|, x ∈ R, where | · | denotes the absolute value |x| := max(x, −x),

x ∈ R

In general, for all x, y in X we have

kxk − kyk

4 Norm Derivatives and Characterizations of Inner Product Spaces

so introducing the mapping d from X × X into R by

d(x, y) := kx − yk, for all x, y in X, we infer that d is a metric induced by the norm k · k, 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 k · k is continuous and the topology induced by the norm is compatible with the vector space operations, i.e., R × X ∋ (α, x) 7→ αx ∈ X and X × X ∋ (x, y) 7→ x+ y ∈ X are continuous in both variables together

X such that ky − xk < r and can be obtained as the x-translation of the ball K0(r) centered at the origin, i.e., Kx(r) = x + K0(r) Analogously, one considers the closed ball

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

sphere S0(1) by SX

When all Cauchy sequences in (X, k · k) 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)])

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

kf(x) − f(y)k = kx − yk, 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 linear space

n

X

i=1

x2

i,

n

X

i=1

|xi|, kxk∨= max{|x1|, , |xn|},







n−1

X

i=1

x2 i

!1/2

, |xn|







for all x = (x1, x2, , xn) in Rn The norm k · ke is the classical Euclidean

natural norm is defined by

kxk∞= sup{|xn| : n ≥ 1},

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

n=1|xn| < ∞ one considers the norm

∞

X

n=1

|xn| for all x = (xn)

6 Norm Derivatives and Characterizations of Inner Product Spaces

(v) The spaces lp,1 < p < ∞

a, b ≥ 0, implies H¨older’s inequality

∞

X

n=1

|xnyn| ≤

∞

X

n=1

|xn|q

X

n=1

|yn|p

!1/p

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 Minkowski’s inequality

∞

X

n=1

|xn+ yn|p

!1/p

≤

∞

X

n=1

|xn|p

!1/p

+

∞

X

n=1

|yn|p

!1/p

, where 1 < p < +∞ Thus the space lp (1 < p < ∞) of infinite sequences (xn) such that P∞

n=1|xn|p< ∞ admits the norm defined by

∞

X

n=1

|xn|p

!1/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

kfk = sup{|f(x)| : x ∈ K}

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

a |f(t)|p dt < ∞

Then one defines the norm

a |f(t)|p dt

!1/p

!1/ p

whenever 1/ p + 1/ q = 1, < p < ∞, and real sequences (xn) and (yn) are such that the right-hand side of (1. 2 .1) converges From (1. 2 .1) ...

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(v) The spaces lp ,1 < p <

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