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[.]

Norm Derivatives and Characterizations of Inner Product Spaces This page intentionally left blank Norm Derivatives and Characterizations of Inner Product Spaces Claudi Alsina Universitat Politècnica de Catalunya, Spain Justyna Sikorska Silesian University, Poland M Santos Tomás Universitat Politècnica de Catalunya, Spain World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TA I P E I • CHENNAI Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library NORM DERIVATIVES AND CHARACTERIZATIONS OF INNER PRODUCT SPACES Copyright © 2010 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN-13 978-981-4287-26-5 ISBN-10 981-4287-26-1 Printed in Singapore 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 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 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 is devoted to the key concepts of the publication: norm derivatives These functionals extend inner products, so many geometrical 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 charv 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, and are devoted to studying heights, perpendicular bisectors and bisectrices in triangles located in normed linear spaces, respectively In doing a detailed study of the basic geometrical properties of these lines and their associated points (orthocenters, circumcenters and incenters), we show a distinguished collection of characterizations of normed linear spaces as inner product spaces Chapter 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± (·, ·) b± (·, ·) BX Bx (r) ◦ dim h·, ·i f −1 inf i.p.s max PA ⊥P ⊥A ⊥B ⊥J ⊥ρ , ⊥ρ R R+ k·k absolute value of a real number difference of two sets angle generalized bisectrix closed unit ball in a normed space closed ball of radious r centered at x composition of functions dimension inner product inverse function of f infimum inner product space maximum minimum metric projection on A Pythagorean orthogonality area orthogonality Birkhoff orthogonality James orthogonality ρ′± orthogonality real line positive half-line norm vii viii Norm Derivatives and Characterizations of Inner Product Spaces ρ′+ , ρ′− ρ′′+ , ρ′′− sgn k · ke k · k+ k · k∨ k · kn X (X, h·, ·i) xy (X, k · k) [x, y] (x, y) hx, yi kxk Sx (r) SX Kx (r) w(·, ·) [x]B k·k [x]ρk·k direction one-sided derivative of the square of the norm direction one-sided derivative of ρ′+ , ρ′− , respectively generalized sign function sgn(x) := x/kxk for x 6= 0, sgn(0) = Euclidean norm taxi-cab norm diamond norm mixed norm generic vector space generic i.p.s line determined by x, y generic real normed space closed segment determined by x, y open closed segment determined by x, y inner product of x, y norm of x sphere of radius r centered at x unit sphere centered at open ball of radius r centered at x vectorial bisectrix segment the Birkhoff’s orthogonal set of x the ρ-orthogonal set of x Contents Preface v Special Notations vii Introduction 1.1 1.2 1.3 1.4 1.5 Historical notes Normed linear spaces Strictly convex normed linear spaces Inner product spaces Orthogonalities in normed linear spaces Norm Derivatives 7 11 15 2.1 2.2 2.3 2.4 2.5 2.6 Norm derivatives: Definition and basic properties Orthogonality relations based on norm derivatives ρ′± -orthogonal transformations On the equivalence of two norm derivatives Norm derivatives and projections in normed linear spaces Norm derivatives and Lagrange’s identity in normed linear spaces 2.7 On some extensions of the norm derivatives 2.8 ρ-orthogonal additivity Norm Derivatives and Heights 15 26 30 35 38 41 45 51 57 3.1 Definition and basic properties 3.2 Characterizations of inner product spaces involving geometrical properties of a height in a triangle ix 57 60 x Norm Derivatives and Characterizations of Inner Product Spaces 3.3 3.4 3.5 3.6 Height functions and classical orthogonalities A new orthogonality relation Orthocenters A characterization of inner product spaces isosceles trapezoid property 3.7 Functional equations of the height transform involving an Perpendicular Bisectors in Normed Spaces Bisectrices in Real Normed Spaces Bisectrices in real normed spaces A new orthogonality relation Functional equation of the bisectrix transform Generalized bisectrices in strictly convex real normed Incenters and generalized bisectrices 91 94 103 4.1 Definitions and basic properties 4.2 A new orthogonality relation 4.3 Relations between perpendicular bisectors and classical orthogonalities 4.4 On the radius of the circumscribed circumference of a triangle 4.5 Circumcenters in a triangle 4.6 Euler line in real normed space 4.7 Functional equation of the perpendicular bisector transform 5.1 5.2 5.3 5.4 5.5 74 81 85 103 106 111 115 117 124 125 131 spaces Areas of Triangles in Normed Spaces 6.1 Definition of four areas of triangles 6.2 Classical properties of the areas and characterizations of inner product spaces 6.3 Equalities between different area functions 6.4 The area orthogonality 131 136 144 149 156 163 163 164 169 172 Appendix A Open Problems 177 Bibliography 179 Index 187 Chapter Introduction 1.1 Historical notes Functional analysis arose from problems on mathematical physics and astronomy where the classical analytical methods were inadequate For example, 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 derivatives 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 spaces, extending the classical notion of the Euclidean spaces Rn , so distance measures could be associated to all kinds of abstract objects This opened up the theory of metric spaces and their future generalizations, extending 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 distances 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 Indeed, 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 Norm Derivatives and Characterizations of Inner Product Spaces Volterra, Fredholm and Hilbert, and made it possible to prove strong results, 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, Kolmogorov 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 characterization (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 characterizations have been found It is also necessary to mention the interest in orthogonally additive mappings developed by Pinsker, Drewnovski, Orlicz, Sunderesan, Gudder, Strawther, Răatz, Szabo, etc (and where in further studies, the second author also made many contributions) as well as solving 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, k · k), the norm derivatives are given for fixed x and y in X by the two expressions lim λ→0± kx + λyk − kxk λ The question [Kă othe (1969)] of when a boundary point of the unit ball has a tangent hyperplane is connected with the differentiability of the norm Introduction at this point [Mazur (1933)], so norm derivatives have been considered in problems looking for smooth conditions (see [Kăothe (1969)], Đ26), but very few characterizations of i.p.s given in terms of norm derivatives were reported in [Amir (1986)] Note that instead of considering the above norm derivatives, it is more convenient to introduce the functionals kx + λyk − kxk kx + λyk2 − kxk2 = kxk · lim λ→0± λ→0± 2λ λ ρ′± (x, y) = lim because when the norm comes from an inner product h·, ·i, we obtain ρ′± (x, y) = hx, yi, i.e., functionals ρ′± are perfect generalizations of inner products Our chief concern in this publication is precisely to see how by virtue of these functionals ρ′± 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, k · k) is called a real normed linear space 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) (ii) (iii) (iv) kxk ≥ for all x in X, kxk = if and only if x = 0, kαxk = |α|kxk for all x in X and α in R, 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 (1.2.1) kxk − kyk ≤ kx − yk ≤ kxk + kyk, 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 The open ball Kx (r) of radius r centered at x consists of all points y in 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 Bx (r) = {y ∈ X : ky − xk ≤ r} = x + B0 (r) The sphere of radius r centered at x will be defined by Sx (r) = {y ∈ X : kx − yk = 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, 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)]) Theorem 1.2.1 Assume that (X, k · k), (Y, k · k) are real normed spaces 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 5 Introduction (i) The space Rn n ≥ being fixed, and with the usual linear structure of Rn , one can consider the following norms: v u n uX x2i , kxke = t i=1 kxk+ = n X i=1 |xi |, kxk∨ = max{|x n |},  |, , |x!  1/2  n−1  X kxkm = max x2i , |xn |   i=1 n for all x = (x1 , x2 , , xn ) in R The norm k · ke is the classical Euclidean norm, k · k+ is the so-called taxi-cab norm, k · k∨ is the diamond norm and k · km is a mixed norm (ii) The spaces c0 , c and l∞ If l∞ denotes the linear space of all bounded real sequences then its natural norm is defined by kxk∞ = 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 (iv) The space l1 In the linear space of real sequences (xn ) such that n=1 considers the norm kxk1 = for all x = (xn ) ∞ P ∞ X n=1 |xn | |xn | < ∞ one Norm Derivatives and Characterizations of Inner Product Spaces (v) The spaces lp , < p < ∞ The classical inequality aα b1−α ≤ αa + (1 − α)b for α in (0,1) and a, b 0, implies Hă olders inequality !1/q !1/p ∞ ∞ X X X q p |xn yn | ≤ |xn | |yn | , (1.2.2) n=1 n=1 n=1 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) one easily derives Minkowski’s inequality !1/p !1/p !1/p ∞ ∞ ∞ X X X p p p ≤ + , |xn + yn | |xn | |yn | n=1 n=1 n=1 p where < p < +∞ Thus the space l (1 < p < ∞) of infinite sequences ∞ P (xn ) such that |xn |p < ∞ admits the norm defined by n=1 ∞ X kxkp = n=1 p |xn | !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 kf k = sup{|f (x)| : x ∈ K} (vii) The Lp -spaces, p ≥ 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 Z b |f (t)|p dt < ∞ a Then one defines the norm kf kp = Z a b p |f (t)| dt !1/p Introduction 1.3 Strictly convex normed linear spaces An important class of real normed linear spaces is formed by the so-called strictly convex spaces Precisely, we quote the following definition (see, e.g., [Kă othe (1969)]): Definition 1.3.1 A real normed linear space (X, k·k) is said to be strictly convex if 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) (ii) (iii) (iv) (v) (vi) SX contains no line segments; Every supporting hyperplane intersects SX in at most one point; Distinct boundary points have distinct supporting planes; If kxk = kyk = and x 6= y then 12 (x + y) < 1; If kx + yk = kxk + kyk and y 6= then x = αy for some α ≥ 0; If x, y in X are linearly independent then kx + yk < kxk + kyk The spaces (Rn , k · ke ), lp and Lp are strictly convex for < p < ∞ and for all n ∈ N, while (Rn , k · k+ ), (Rn , k · k∨ ), (Rn , k · km ), c0 , l1 , l∞ , L1 , for n > 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 h·, ·i on X × X that satisfies the following four properties for all x, y, z in X and α in R: (i) (ii) (iii) (iv) hx, xi is nonnegative and hx, xi = if and only if x = 0, hx, y + zi = hx, yi + hx, zi, hx, αyi = αhx, yi, hx, yi = hy, xi An inner product h·, ·i defined on X × X induces the norm p kxk := hx, xi, x ∈ X so 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, k · k) is a Banach space and k · k derives from an inner product h·, ·i, then (X, h·, ·i) is called a Hilbert space One can show that any i.p.s is strictly convex 8 Norm Derivatives and Characterizations of Inner Product Spaces In the Hilbert space (Rn , k·ke ) one considers the standard inner product hx, yi = n X xj yj j=1 for all vectors x = (x1 , , xn ), y = (y1 , , yn ) in Rn In the space L2 of continuous real-valued functions on [a, b] such that Z b |f (t)|2 dt < ∞, a one may consider the inner product Z b hf, gi = f (t)g(t)dt a In the space l one defines its inner product structure by means of the expression hx, yi = ∞ X xn yn n=1 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 Neumann (1935)] states the following criterium for checking when the norm derives from an inner product Theorem 1.4.1 Let (X, k · k) be a real normed linear space Then k · k derives from an inner product if and only if the parallelogram law holds, i.e., kx + yk2 + kx − yk2 = 2kxk2 + 2kyk2 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 Introduction 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)] Theorem 1.4.2 Let (X, k · k) be a real normed linear space Then k · k derives from an inner product if and only if kx + yk2 + kx − yk2 ∼ 2kxk2 + 2kyk2 for all x, y ∈ X, where ∼ stands either for ≤ or ≥ Theorem 1.4.3 Let (X, k · k) be a real normed linear space Then k · k derives from an inner product if and only if ku + vk2 + ku − vk2 ∼ for all u, v ∈ SX , where ∼ stands for one of the signs =, ≤ or ≥ Theorem 1.4.4 [Ben´ıtez and del Rio (1984)] Let (X, k · k) be a real normed linear space Then k · k derives from an inner product if and only if for all u, v in SX there exist α, β 6= such that kαµ + βvk2 + kαu − βvk2 ∼ 2(α2 + β ), where ∼ stands for =, ≤ or ≥ When the norm k · k derives from an inner product h·, ·i then the inner product can be obtained from the norm by means of the polarization identity hx, yi = (kx + yk2 − kx − yk2 ) In our considerations, we will still require another classical characterization of inner product spaces Theorem 1.4.5 Let (X, k · k) be a real normed linear space of dimension greater than or equal to 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, h·, ·i) are said to be orthogonal if hx, yi = A collection {xi } of vectors in X is called an orthonormal set if hxi , xi i = and hxi , xj i = and for all positive integers i, j and i 6= j  ... real normed space 4.7 Functional equation of the perpendicular bisector transform 5 .1 5.2 5.3 5.4 5.5 74 81 85 10 3 10 6 11 1 11 5 11 7 12 4 12 5 13 1 spaces Areas of Triangles.. .Norm Derivatives and Characterizations of Inner Product Spaces This page intentionally left blank Norm Derivatives and Characterizations of Inner Product Spaces Claudi Alsina... orthogonality 13 1 13 6 14 4 14 9 15 6 16 3 16 3 16 4 16 9 17 2 Appendix A Open Problems 17 7 Bibliography 17 9 Index 18 7 Chapter Introduction 1. 1 Historical notes Functional analysis