Banach-Stone theorems for vector valued functions on completely regular spaces

We obtain several Banach-Stone type theorems for vector-valued functions in this paper. Let X, Y be realcompact or metric spaces, E, F locally convex spaces, and phi a bijective linear map from C(X, E) onto C(Y, F). If phi preserves zero set containments, i.e., z(f) subset of z(g) <-> z(phi(f)) subset of z(phi(g)), for all f, g is an element of C(X, E), then X is homeomorphic to V. and phi is a weighted composition operator. The above conclusion also holds if we assume a seemingly weaker condition that phi preserves nonvanishing functions, i.e., z(f) = empty set <-> z(phi f) = empty set, for all f is an element of C(X, E). These two results are special cases of the theorems in a very general setting in this paper, covering bounded continuous vector-valued functions on general completely regular spaces, and uniformly continuous vector-valued functions on metric spaces. Our results extend and generalize many recent ones. Crown Copyright (C) 2012 Published by Elsevier Inc. All rights reserved.