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).
Author(s): Claudi Alsina, Justyna Sikorska, M. Santos Tomas
Publisher: World Scientific Publishing Company
Year: 2010
Language: English
Pages: 188
Preface......Page 6
Special Notations......Page 8
Contents......Page 10
1.1 Historical notes......Page 12
1.2 Normed linear spaces......Page 14
1.4 Inner product spaces......Page 18
1.5 Orthogonalities in normed linear spaces......Page 22
2.1 Norm derivatives: Definition and basic properties......Page 26
2.2 Orthogonality relations based on norm derivatives......Page 37
2.3 p′±-orthogonal transformations......Page 41
2.4 On the equivalence of two norm derivatives......Page 46
2.5 Norm derivatives and projections in normed linear spaces......Page 49
2.6 Norm derivatives and Lagrange’s identity in normed linear spaces......Page 52
2.7 On some extensions of the norm derivatives......Page 56
2.8 p-orthogonal additivity......Page 62
3.1 Definition and basic properties......Page 68
3.2 Characterizations of inner product spaces involving geometrical properties of a height in a triangle......Page 71
3.3 Height functions and classical orthogonalities......Page 85
3.4 A new orthogonality relation......Page 92
3.5 Orthocenters......Page 96
3.6 A characterization of inner product spaces involving an isosceles trapezoid property......Page 102
3.7 Functional equations of the height transform......Page 105
4.1 Definitions and basic properties......Page 114
4.2 A new orthogonality relation......Page 117
4.3 Relations between perpendicular bisectors and classical orthogonalities......Page 122
4.4 On the radius of the circumscribed circumference of a triangle......Page 126
4.5 Circumcenters in a triangle......Page 128
4.6 Euler line in real normed space......Page 135
4.7 Functional equation of the perpendicular bisector transform......Page 136
5.1 Bisectrices in real normed spaces......Page 142
5.2 A new orthogonality relation......Page 147
5.3 Functional equation of the bisectrix transform......Page 155
5.4 Generalized bisectrices in strictly convex real normed spaces......Page 160
5.5 Incenters and generalized bisectrices......Page 167
6.1 Definition of four areas of triangles......Page 174
6.2 Classical properties of the areas and characterizations of inner product spaces......Page 175
6.3 Equalities between different area functions......Page 180
6.4 The area orthogonality......Page 183
Appendix A: Open Problems......Page 187
Bibliography......Page 189
Index......Page 197
