Weak compactness in Lipschitz-free spaces over superreflexive spaces

TL;DR

This paper demonstrates that Lipschitz-free spaces over superreflexive Banach spaces exhibit strong compactness properties similar to $L_1$, including weak Banach-Saks property and superreflexivity of certain subspaces.

Contribution

It establishes that weakly precompact subsets of these Lipschitz-free spaces are super weakly compact, extending understanding of their structure and properties.

Findings

Weakly precompact subsets are super weakly compact

Lipschitz-free spaces have the weak Banach-Saks property

Subspaces with nontrivial type are superreflexive

Abstract

We show that the Lipschitz-free space $\mathcal{F}(X)$ over a superreflexive Banach space $X$ has the property that every weakly precompact subset of $\mathcal{F}(X)$ is relatively super weakly compact, showing that this space "behaves like $L_1$" in this context. As consequences we show that $\mathcal{F}(X)$ enjoys the weak Banach-Saks property and that every subspace of $\mathcal{F}(X)$ with nontrivial type is superreflexive. Further, weakly compact subsets of $\mathcal{F}(X)$ are super weakly compact and hence have many strong properties. To prove the result, we use a modification of the proof of weak sequential completeness of $\mathcal{F}(X)$ by Kochanek and Perneck\'a and an appropriate version of compact reduction in the spirit of Aliaga, No\^us, Petitjean and Proch\'azka.