A note on nonseparable Lipschitz-free spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913300485242880 |
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| author | Aliaga, Ramón J. Grelier, Guillaume Procházka, Antonín |
| author_facet | Aliaga, Ramón J. Grelier, Guillaume Procházka, Antonín |
| contents | We prove that several classical Banach space properties are equivalent to separability for the class of Lipschitz-free spaces, including Corson's property ($\mathcal{C}$), Talponen's Countable Separation Property, or being a Gâteaux differentiability space. On the other hand, we single out more general properties where this equivalence fails. In particular, the question whether the duals of non-separable Lipschitz-free spaces have a weak$^*$ sequentially compact ball is undecidable in ZFC. Finally, we provide an example of a nonseparable dual Lipschitz-free space that fails the Radon-Nikodým property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14678 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on nonseparable Lipschitz-free spaces Aliaga, Ramón J. Grelier, Guillaume Procházka, Antonín Functional Analysis 46B20 (Primary) 46B26, 46E15 (Secondary) We prove that several classical Banach space properties are equivalent to separability for the class of Lipschitz-free spaces, including Corson's property ($\mathcal{C}$), Talponen's Countable Separation Property, or being a Gâteaux differentiability space. On the other hand, we single out more general properties where this equivalence fails. In particular, the question whether the duals of non-separable Lipschitz-free spaces have a weak$^*$ sequentially compact ball is undecidable in ZFC. Finally, we provide an example of a nonseparable dual Lipschitz-free space that fails the Radon-Nikodým property. |
| title | A note on nonseparable Lipschitz-free spaces |
| topic | Functional Analysis 46B20 (Primary) 46B26, 46E15 (Secondary) |
| url | https://arxiv.org/abs/2312.14678 |