Characterization of inner product spaces

TL;DR

This paper characterizes real inner product spaces of dimension greater than two through properties of best coapproximations and orthonormal bases, providing new insights into the structure of normed spaces.

Contribution

It establishes a new characterization of inner product spaces based on coapproximation and orthonormal bases, and verifies this for spaces with p-norms.

Findings

Best coapproximation coincides with best approximation in inner product spaces.

Characterization of inner product spaces via coapproximation properties.

Verification of the conjecture for r^n,_p spaces.

Abstract

We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterizes a real inner product space of dimension $ > 2 $. We conjecture that a finite dimensional real smooth normed space of dimension $ >2 $ is an inner product space iff given any element on the unit sphere there exists a strongly orthonormal Hamel basis in the sense of Birkhoff-James containing that element. This is substantiated by our result on the spaces $(R^n,\|.\|_p).$