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arXiv · 2408.01135

Weak compactness in Lipschitz-free spaces over superreflexive spaces

Zdeněk Silber
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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.

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Zdeněk Silber. 2024-08-02. Weak compactness in Lipschitz-free spaces over superreflexive spaces. https://arxiv.org/abs/2408.01135

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