Approximate Isometries of the Space of Continuous Functions

Abstract

A transformation of a metric space which changes distances by less than a certain fixed positive number e is called an approximate isometry, or an eisometry. In a preceding paper2 it was shown that an e-isometry taking Hilbert space into itself could be uniformly approximated within a distance of ke by a true isometry, where k was a positive constant not greater than 10. The object of the present paper is to establish a similar theorem for transformations of the space of continuous functions over a compact metric space. According to a theorem of Banach,8 every isometry between the normed spaces E and E' of continuous, real valued functions over the compact metric spaces K and K', respectively, is generated by a homeomorphism between K and K', and conversely. This result suggests a method of attack on our stability problem. The first step is to prove the existence of an isometry between the spaces E and E', having given the approximate isometry between these spaces, and this can be done by showing that the underlying spaces K and K' are homeomorphic. However, the method used by Banach in establishing the correspondence between the points of K and K', which was based on a theory of peak functions, seemed rather difficult to apply to our problem. Therefore a different method, dealing with the hyperplane of all functions having the same value at a given point of K, is employed in the present paper. The norms in the spaces E and E' will be defined in the usual way, as the maximum of the absolute value of the function in question. A transformation T(f) taking E into E' will be called an e-isometry if II T(f) T(g) II If gII I 0 there exists a point q of K' and a redl number c such that T(M(p b, a)) C M'(q, c, a + 3e/2). PROOF. Without loss of generality, we may assume that the metrics in K and K' have been chosen so that the diameter of each of these spaces is one. We shall assume for the present that a > 0, and consider the case a = 0 later. Putfn(x) = b + 3n np(x,p), gn(X) = b 3n + np(x, p), for n = 1, 2, 3, * * and x in K, where p(x, p) denotes the distance from x to p. Consider the spheres Sn = [kp; II (p-fIn I _ 3n + a], R. = [(p; I I -gn I I _ 3n + a] in the space E