This is an announcement for the paper "A characterization of inner product spaces" by Mohammad Sal Moslehian and John M. Rassias.

Abstract: In this paper we present a new criterion on characterization of real inner product spaces. We conclude that a real normed space $(X, |\cdot|)$ is an inner product space if $$\sum_{\varepsilon_i \in {-1,1}} \left|x_1 + \sum_{i=2}^k\varepsilon_ix_i\right|^2=\sum_{\varepsilon_i \in {-1,1}} \left(|x_1| + \sum_{i=2}^k\varepsilon_i|x_i|\right)^2,,$$ for some positive integer $k\geq 2$ and all $x_1, \ldots, x_k \in X$. Conversely, if $(X, |\cdot|)$ is an inner product space, then the equality above holds for all $k\geq 2$ and all $x_1, \ldots, x_k \in X$.

Archive classification: math.FA math.CA

Mathematics Subject Classification: Primary 46C15, Secondary 46B20, 46C05

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

Submitted from: moslehian@ferdowsi.um.ac.ir

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1009.0079

or

http://arXiv.org/abs/1009.0079