A characterization of inner product spaces

Abstract

In this paper we present a new criterion on characterization of real inner product spaces. We conclude that a real normed space $(X,\Vert ...\Vert )$ is an inner product space if $$\sum _{{\epsilon }_i\in \{-1,1\}}\Vert x_1+\sum _{i=2}^k{\epsilon }_i{x}_i{\Vert }^2=\sum _{{\epsilon }_i\in \{-1,1\}}(\Vert x_1\Vert +\sum _{i=2}^k{\epsilon }_i\Vert x_i\Vert {)}^2,$$ for some positive integer $k\ge 2$ and all $x_1,...,x_k\in X$. Conversely, if $(X,\Vert ...\Vert )$ is an inner product space, then the equality above holds for all $k\ge 2$ and all $x_1,...,x_k\in X$.

8 Pages, to appear in Kochi J. Math. (Japan)