A characterization of inner product spaces via norming vectors

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.

v2 : modified title, added reference to Sain-Paul ; to appear in Canadian Mathemathical Bulletin