Norm Derivatives and Characterizations of Inner Product Spaces

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More About This Textbook


The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordan-von Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces.

This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications).

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Product Details

  • ISBN-13: 9789814287265
  • Publisher: World Scientific Publishing Company, Incorporated
  • Publication date: 11/28/2009
  • Pages: 200
  • Product dimensions: 5.90 (w) x 9.00 (h) x 0.80 (d)

Table of Contents

Preface v

Special Notations vii

1 Introduction 1

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'&Plus-minus;-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 geometrical properties of a height in a triangle 60

3.3 Height functions and classical orthogonalities 74

3.4 A new orthogonality relation 81

3.5 Orthocenters 85

3.6 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

4.2 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 Bisectrices in Real Normed Spaces 131

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 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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