An Approximate Version of the Jordan von Neumann Theorem for Finite Dimensional Real Normed Spaces

TL;DR

This paper establishes that finite-dimensional real normed spaces approximately satisfying the parallelogram law are close to Euclidean spaces, with the closeness quantified by the von Neumann Jordan constant and its growth with dimension.

Contribution

It formulates an approximate version of the Jordan von Neumann theorem for finite-dimensional spaces and analyzes how the approximation quality deteriorates as dimension increases.

Findings

Small Jordan constant implies near Euclidean structure.

Banach-Mazur distance is bounded by a linear function of the Jordan constant.

Dimension affects the approximation quality quadratically.

Abstract

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratically with n.