Approximate Isometries on Finite Dimensional Banach Spaces by Richard

A map T: Ej —► E2 (E|, E2 Banach spaces) is an e-isometry if III T(X) T(Y)\\ \\X Y\\ I < e whenever X, Ye Ex. The problem of uniformly approximating such maps by isometries was first raised by Hyers and Ulam in 1945 and subsequently studied for special infinite dimensional Banach spaces. This question is here broached for the class of finite dimensional Banach spaces. The only positive homogeneous candidate isometry U approximating a given e-isometry T is defined by the formal limit U(X) = lim^^r T(rX). It is shown that, whenever T: E —► E is a surjective e-isometry and E is a finite dimensional Banach space for which the set of extreme points of the unit ball is totally disconnected, then this limit exists. When E —t x (= fc-dimensional Pj) a uniform bound of uniform approximation is obtained for surjective c-isometries by isometries; this bound 3 varies linearly in e and with k . 1. The form in which Hyers and Ulam [6] considered the e-isometry question is: (1.1) Does there exist a constant K depending only on Ej and E2 with the following property: For each e > 0 and surjective e-isometry T: Ej —► E2 there is an isometry U: Ex —► E2 with \\T(X) U(X)\\ < Ke for each X in E,? They observed that the assumption that T be surjective is essential and answered (1.1) affirmatively when Ex = E2 = Hubert space [6]. D. G. Bourgin [2] showed more generally that (1.1) holds whenever Ej and E2 belong to a class of uniformly convex Banach spaces including the Lp(X, 2, u) spaces 1 < p < °°. A subsequent paper of Hyers and Ulam [7] gave a positive answer for E¿ = C(Df), i = 1, 2 (the Banach spaces of continuous functions on the compact Hausdorff spaces Df with the sup norm), provided Pis a homeomorphism. This study was continued Received by the editors March 19, 1974. AMS {MOS) subject classifications (1970). Primary 46A05; Secondary 46B99.