Characterizing of Inner Product Spaces by the Mapping $n_{x,y}$

TL;DR

This paper provides a new characterization of inner product spaces by analyzing the symmetry of the mappings defined by the norms of linear combinations of vectors.

Contribution

It introduces a simple criterion based on the comparison of mappings n_{x,y} and n_{y,x} to identify inner product spaces.

Findings

Inner product spaces can be characterized by the symmetry of the mappings n_{x,y} and n_{y,x}

The mapping comparison offers a straightforward test for inner product space structure

The approach simplifies the identification of inner product spaces in normed linear spaces

Abstract

For the vectors $x$ and $y$ in a normed linear spaces $X$, the mapping $n_{x,y}: \mathbb{R}\to \mathbb{R}$ is defined by $n_{x,y}(t)=\|x+ty\|$. In this note, comparing the mappings $n_{x,y}$ and $n_{y,x}$ we obtain a simple and useful characterization of inner product spaces.