Banach Spaces of Lipschitz Functions

Abstract

In this chapter we introduce several Banach spaces of Lipschitz functions (Lipschitz functions vanishing at a fixed point, bounded Lipschitz functions, little Lipschitz functions) on a metric space and present some of their properties. A detailed study of free Lipschitz spaces is carried out, including several ways to introduce them and duality results. The study of Monge–Kantorovich and Hanin norms is tightly connected with Lipschitz spaces, mainly via the weak convergence of probability measures, a topic treated in Sects.8.4 and 8.5. Compactness and weak compactness properties of Lipschitz operators on Banach spaces and of compositions operators on spaces of Lipschitz functions are also studied, emphasizing the key role played by the Lipschitz free Banach spaces. Another theme presented here is the Bishop–Phelps property for Lipschitz functions, meaning density results for Lipschitz functions that attain their norms. Finally, applications to best approximation in metric spaces and in metric linear spaces X are given in the last section (Sect. 8.9) of this chapter, showing how results from the linear theory can be transposed to this situation, by using as a dual space the space of Lipschitz functions defined on X.