A characterization of inner product spaces via norming vectors

TL;DR

This paper provides a new proof characterizing finite-dimensional inner product spaces through the properties of norming vectors, extending previous results from real to complex scalar fields.

Contribution

It offers an alternative proof of a characterization theorem for inner product spaces that also applies to complex scalars, broadening the scope of the original result.

Findings

The set of norming vectors forms a linear subspace in inner product spaces.

The characterization holds for both real and complex finite-dimensional spaces.

A new proof technique is introduced that extends previous real scalar results.

Abstract

A finite-dimensional normed space is an inner product space if and only if the set of norming vectors of any endomorphism is a linear subspace. This theorem was proved by Sain and Paul for real scalars. In this paper, we give a different proof which also extends to the case of complex scalars.