Spaces Of Lipschitz Functions On Banach Spaces

TL;DR

This paper explores the structure and properties of Lipschitz functions on Banach spaces, extending classical theorems like James' theorem to analyze support sets and their implications for the geometry of Banach spaces.

Contribution

It generalizes James' theorem to support sets in Banach spaces and investigates the structure of Lipschitz functions and their support sets in this context.

Findings

Support sets can be characterized in terms of weak compactness.

The structure of support sets influences the separability of dual spaces.

Applications include new insights into the geometry of Banach spaces.

Abstract

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a weakly compact set. Actually, this result is significantly stronger than this statement; indeed, the proof can be used to obtain other surprising results. For example, suppose that $X$ is a separable Banach space and $S$ is a norm separable subset of the unit ball of $X^*$ such that for each $x \in X$ there exists $x^* \in S$ such that $x^*(x) = \|x\|$ then $X^*$ is itself norm separable . If we call $S$ a support set, in this case, with respect to the entire space $X$, one can ask questions about the size and structure of a support set, a support set not only with respect to $X$ itself but perhaps with respect to some other subset of $X$@. We analyze one particular case of this as well as give some applications.