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Characterization of inner product spaces

Debmalya Sain, Kallol Paul, Lokenath Debnath

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 $(R^n,\|.\|_p).$

Characterization of inner product 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 . We conjecture that a finite dimensional real smooth normed space of dimension 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

Paper Structure

This paper contains 2 sections, 5 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Strongly orthonormal Hamel basis and best approximation & best coapproximation

Key Result

Theorem 2.2

Suppose X is a strictly convex normed linear space and $S= \{ x_1,x_2, \ldots, x_n\}$ is a strongly orthonormal set in the sense of Birkhoff-James. Let $x_{n+1} \in S_X - span~\{x_1,x_2, \ldots, x_n\}.$ Let $w \in span~\{ x_1,x_2, \ldots, x_n\}$ be such that $\{ x_1,x_2, \ldots, x_n, \frac{x_{n+1} -

Theorems & Definitions (11)

  • Remark 2.1
  • Theorem 2.2
  • Example 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Example 2.9
  • Example 2.10
  • ...and 1 more