The ZOO of combinatorial Banach spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910395858419712 |
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| author | Borodulin-Nadzieja, Piotr Farkas, Barnabás Jachimek, Sebastian Pelczar-Barwacz, Anna |
| author_facet | Borodulin-Nadzieja, Piotr Farkas, Barnabás Jachimek, Sebastian Pelczar-Barwacz, Anna |
| contents | We study Banach spaces induced by families of finite sets in the most natural (Schreier-like) way, that is, we consider the completion $X_\mc{F}$ of $c_{00}$ with respect to the norm $\sup\{\sum_{k\in F}|x(k)|:F\in\mc{F}\}$ where $\mc{F}$ is an arbitrary (not necessarily compact) family of finite sets covering $\mbb{N}$.
Among other results, we discuss the following:
(1) Structure theorems bonding the combinatorics of $\mc{F}$ and the geometry of $X_\mc{F}$ including possible characterizations and variants of the Schur property, $\ell_1$-saturation, and the lack of copies of $c_0$ in $X_\mc{F}$.
(2) A plethora of examples including a relatively simple $\ell_1$-saturated combinatorial space which does not satisfy the Schur property, as well as a new presentation of Pełczyński's universal space.
(3) The complexity of the family $\{H\subseteq\NN:X_{\mc{F}\upharpoonright H}$ does not contain $c_0\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The ZOO of combinatorial Banach spaces Borodulin-Nadzieja, Piotr Farkas, Barnabás Jachimek, Sebastian Pelczar-Barwacz, Anna Functional Analysis Logic We study Banach spaces induced by families of finite sets in the most natural (Schreier-like) way, that is, we consider the completion $X_\mc{F}$ of $c_{00}$ with respect to the norm $\sup\{\sum_{k\in F}|x(k)|:F\in\mc{F}\}$ where $\mc{F}$ is an arbitrary (not necessarily compact) family of finite sets covering $\mbb{N}$. Among other results, we discuss the following: (1) Structure theorems bonding the combinatorics of $\mc{F}$ and the geometry of $X_\mc{F}$ including possible characterizations and variants of the Schur property, $\ell_1$-saturation, and the lack of copies of $c_0$ in $X_\mc{F}$. (2) A plethora of examples including a relatively simple $\ell_1$-saturated combinatorial space which does not satisfy the Schur property, as well as a new presentation of Pełczyński's universal space. (3) The complexity of the family $\{H\subseteq\NN:X_{\mc{F}\upharpoonright H}$ does not contain $c_0\}$. |
| title | The ZOO of combinatorial Banach spaces |
| topic | Functional Analysis Logic |
| url | https://arxiv.org/abs/2404.01733 |