An approximate version of the Jordan von Neumann theorem for finite-dimensional real normed spaces

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 ε + 1 for X yields a small Banach-Mazur distance with R n , d(X, R n ) ≤ 1 + β n ε + O(ε 2 ). Finally, we examine how this estimate worsens as n = dim(X ) increases, with the conclusion that β n grows quadratically with n.