Operator Characterizations of Inner Product Spaces

Abstract

As we have seen, if on a normed linear space X there exist certain relations among finite sets of elements in X then this forces the space to be an inner product space. Sometimes this fact may be interpreted to state that certain functions defined on the space X have some very special properties. In the first part of this chapter we present further characterizations of inner product spaces along these lines. Our approach is that we first present certain characterizations using nonlinear mappings. We begin with a characterization of inner product spaces obtained by I. J. Schoenberg using a class of functions related to negative definite functions. Secondly, we present certain inequalities on Banach spaces, including the characterization of inner product spaces using a sharp form of an inequality. The case of nonexpansive mappings and their connections with the characterization of inner product spaces, as well as with the m-accretive and maximal accretive mappings, is discussed further. Here we present some of the classical results of, for example, K. Kirszbraun and F. Valentine and also some recent ones by B. Grunbaum, S. Schonbeck, etc.