Small isomorphisms of C(X, E) spaces

Abstract

A linear map T between two Banach spaces A and B is called ε-isometry if (1 - ε)‖f‖ ≤ ‖Tf‖ ≤ (1 + ε)‖f‖, for any f ∈ A. In the paper we investigate injective and surjective ε-isometries between Banach spaces of continuous E-valued functions. We prove that, under some geometrical assumptions on the Banach space E, any such ε-isometry is induced by a continuous function between the corresponding compact Hausdorff spaces. We discuss also the question whether such an ε-isometry has to be just a small perturbation of an isometry. © 1989 by Pacific Journal of Mathematics.