A characterization of inner product spaces

TL;DR

This paper introduces a new criterion involving sum of squared norms over sign combinations to characterize real inner product spaces, providing a novel mathematical condition for identifying such spaces.

Contribution

It presents a new necessary and sufficient condition based on norm sums over sign variations to characterize real inner product spaces.

Findings

The criterion characterizes inner product spaces precisely.

The condition holds for all k ≥ 2 in inner product spaces.

The criterion is both necessary and sufficient.

Abstract

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