On the spaces dual to combinatorial Banach spaces
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913654985719808 |
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| author | Borodulin-Nadzieja, Piotr Jachimek, Sebastian Pelczar-Barwacz, Anna |
| author_facet | Borodulin-Nadzieja, Piotr Jachimek, Sebastian Pelczar-Barwacz, Anna |
| contents | We present quasi-Banach spaces which are closely related to the duals of combinatorial Banach spaces. More precisely, for a compact family $\mathcal{F}$ of finite subsets of $ω$ we define a quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ whose Banach envelope is the dual norm for the combinatorial space generated by $\mathcal{F}$. Such quasi-norms seem to be much easier to handle than the dual norms and yet the quasi-Banach spaces induced by them share many properties with the dual spaces. We show that the quasi-Banach spaces induced by large families (in the sense of Lopez-Abad and Todorcevic) are $\ell_1$-saturated and do not have the Schur property. In particular, this holds for the Schreier families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_10710 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the spaces dual to combinatorial Banach spaces Borodulin-Nadzieja, Piotr Jachimek, Sebastian Pelczar-Barwacz, Anna Functional Analysis 46B20, 46B45 We present quasi-Banach spaces which are closely related to the duals of combinatorial Banach spaces. More precisely, for a compact family $\mathcal{F}$ of finite subsets of $ω$ we define a quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ whose Banach envelope is the dual norm for the combinatorial space generated by $\mathcal{F}$. Such quasi-norms seem to be much easier to handle than the dual norms and yet the quasi-Banach spaces induced by them share many properties with the dual spaces. We show that the quasi-Banach spaces induced by large families (in the sense of Lopez-Abad and Todorcevic) are $\ell_1$-saturated and do not have the Schur property. In particular, this holds for the Schreier families. |
| title | On the spaces dual to combinatorial Banach spaces |
| topic | Functional Analysis 46B20, 46B45 |
| url | https://arxiv.org/abs/2210.10710 |