On free bases of Banach spaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916236961513472 |
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| author | Pernecká, E. Spěvák, J. |
| author_facet | Pernecká, E. Spěvák, J. |
| contents | We call a closed subset M of a Banach space X a free basis of X if it contains the null vector and every Lipschitz map from M to a Banach space Y, which preserves the null vectors can be uniquely extended to a bounded linear map from X to Y. We then say that two complete metric spaces M and N are Mol-equivalent if they admit bi-Lipschitz copies M' and N', respectively that are free bases of a common Banach space satisfying span M'=span N'.
In this note, we compare Mol-equivalence with some other natural equivalences on the class of complete metric spaces. The main result states that Mol-equivalent spaces must have the same Čech-Lebesgue covering dimension. In combination with the work of Godard, this implies that two complete metric spaces with isomorphic Lipschitz-free spaces need not be Mol-equivalent. Also, there exist non-homeomorphic Mol-equivalent metric spaces, and, in contrast with the covering dimension, the metric Assouad dimension is not preserved by Mol-equivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03556 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On free bases of Banach spaces Pernecká, E. Spěvák, J. Functional Analysis General Topology 46B03, 54E35, 54F45 We call a closed subset M of a Banach space X a free basis of X if it contains the null vector and every Lipschitz map from M to a Banach space Y, which preserves the null vectors can be uniquely extended to a bounded linear map from X to Y. We then say that two complete metric spaces M and N are Mol-equivalent if they admit bi-Lipschitz copies M' and N', respectively that are free bases of a common Banach space satisfying span M'=span N'. In this note, we compare Mol-equivalence with some other natural equivalences on the class of complete metric spaces. The main result states that Mol-equivalent spaces must have the same Čech-Lebesgue covering dimension. In combination with the work of Godard, this implies that two complete metric spaces with isomorphic Lipschitz-free spaces need not be Mol-equivalent. Also, there exist non-homeomorphic Mol-equivalent metric spaces, and, in contrast with the covering dimension, the metric Assouad dimension is not preserved by Mol-equivalence. |
| title | On free bases of Banach spaces |
| topic | Functional Analysis General Topology 46B03, 54E35, 54F45 |
| url | https://arxiv.org/abs/2405.03556 |