How Lipschitz Functions Characterize the Underlying Metric Spaces

Abstract

Abstract. Let X, Y be metric spaces and E, F be Banach spaces. Suppose that both X, Y are realcompact, or both E, F are realcompact. The zero set of a vector-valued function f is denoted by z( f ). A linear bijection T between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions ifrespectively. Every zero-set containment preserver, and every nonvanishing function preserver when dim E = dim F < +∞, is a weighted composition operator (T f )(y) = J y ( f (τ (y))). We show that the map τ : Y → X is a locally (little) Lipschitz homeomorphism.