Weak compactness in Lipschitz-free spaces over superreflexive spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913456298393600 |
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| author | Silber, Zdeněk |
| author_facet | Silber, Zdeněk |
| contents | 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 Pernecká and an appropriate version of compact reduction in the spirit of Aliaga, Noûs, Petitjean and Procházka. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01135 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak compactness in Lipschitz-free spaces over superreflexive spaces Silber, Zdeněk Functional Analysis 46B03, 46B20, 46B50 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 Pernecká and an appropriate version of compact reduction in the spirit of Aliaga, Noûs, Petitjean and Procházka. |
| title | Weak compactness in Lipschitz-free spaces over superreflexive spaces |
| topic | Functional Analysis 46B03, 46B20, 46B50 |
| url | https://arxiv.org/abs/2408.01135 |