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 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.