مشتقات نرم و مشخصه‌یابی فضاهای ضرب داخلی ۲۰۱۰
Norm Derivatives and Characterizations of Inner Product Spaces 2010

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). Sample Chapter(s). Chapter 1: Introduction (160 KB). Contents: Norm Derivatives; Norm Derivatives and Heights; Perpendicular Bisectors in Normed Spaces; Bisectrices in Real Normed Spaces; Areas of Triangles in Normed Spaces. Readership: Advanced undergraduates, graduate students, mathematicians and scientists interested in Banach and Hilbert spaces, functional equations and their applications, inequalities and geometry in normed spaces and inner product spaces.