On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911087106981888 |
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| author | Albiac, Fernando Ansorena, Jose L. |
| author_facet | Albiac, Fernando Ansorena, Jose L. |
| contents | With the aim to better understand the intricate geometry of the class of Lipschitz free $p$-spaces $\mathcal{F}_p(\mathcal{M})$ when $0<p<1$, in this note we study their Banach envelopes and prove that if $0<p<1$ and $ \mathcal{M}$ is a metric space then the Banach envelope map of $\mathcal{F}_p(\mathcal{M})$ is one-to-one, thus solving in the positive a problem raised by Kalton in [F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of $ \mathcal{F}_p(\mathcal{M})$ separates the points of $\mathcal{F}_p(\mathcal{M})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$ Albiac, Fernando Ansorena, Jose L. Functional Analysis 26A16 (Primary) 46A16, 46A20, 46B80, 46B20 (Secondary) With the aim to better understand the intricate geometry of the class of Lipschitz free $p$-spaces $\mathcal{F}_p(\mathcal{M})$ when $0<p<1$, in this note we study their Banach envelopes and prove that if $0<p<1$ and $ \mathcal{M}$ is a metric space then the Banach envelope map of $\mathcal{F}_p(\mathcal{M})$ is one-to-one, thus solving in the positive a problem raised by Kalton in [F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of $ \mathcal{F}_p(\mathcal{M})$ separates the points of $\mathcal{F}_p(\mathcal{M})$. |
| title | On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$ |
| topic | Functional Analysis 26A16 (Primary) 46A16, 46A20, 46B80, 46B20 (Secondary) |
| url | https://arxiv.org/abs/2508.00490 |