A note on nonlinear isometries between vector-valued function spaces

Abstract

ABSTRACTLet X and Y be compact Hausdorff spaces, E and F be Banach spaces over or and let A and B be subspaces of and , respectively. In this paper, we investigate the general form of isometries (not necessarily linear) T from A onto B. If F is strictly convex, then there exist a subset of Y , a continuous function onto the set of strong boundary points of A and a family of real-linear operators from E to F with such that In particular, we get some generalizations of the vector-valued Banach–Stone theorem and a generalization of a result of Cambern. We also give a similar result when F lacks the strict convexity and its unit sphere contains a singleton as a maximal convex subset.