How Lipschitz Functions Characterize the Underlying Metric Spaces

Abstract

AbstractLetXandYbe metric spaces andE,Fbe Banach spaces. Suppose that bothXandYare realcompact, or bothE, Fare realcompact. The zero set of a vector-valued function ƒ is denoted byz(ƒ). 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 dimE= dimF<+∞, is a weighted composition operator. We show that the mapτ:Y→Xis a locally (little) Lipschitz homeomorphism.