Small-dimensional normed barrelled 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_ | 1866912110518206464 |
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| author | Brian, Will Stuart, Christopher |
| author_facet | Brian, Will Stuart, Christopher |
| contents | We prove that every separable Banach space has a barrelled subspace with algebraic dimension $\mathrm{non}(\mathcal M)$, which denotes the smallest cardinality of a non-meager subset of $\mathbb R$. This strengthens a theorem of Sobota. More generally, we prove that every Banach space with density character $κ$ contains a barrelled subspace with algebraic dimension $\mathrm{cf}[κ]^ω\cdot \mathrm{non}(\mathcal M)$, and in particular it is consistent with $\mathsf{ZFC}$ that every Banach space with density character $<\!\mathfrak{c}$ has a barrelled subspace with dimension $<\!\mathfrak{c}$.
We also prove that if the dual of a Banach space contains either $c_0$ or $\ell^p$ for some $p \geq 1$, then that space does not have a barrelled subspace with dimension $<\!\mathrm{cov}(\mathcal N)$, which denotes the smallest cardinality of a collection of Lebesgue null sets covering $\mathbb R$. In particular, it is consistent with $\mathsf{ZFC}$ that no classical Banach spaces contain barrelled subspaces with dimension $\mathfrak{b}$. This partly answers a question of Sánchez Ruiz and Saxon. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_13970 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small-dimensional normed barrelled spaces Brian, Will Stuart, Christopher Functional Analysis Logic We prove that every separable Banach space has a barrelled subspace with algebraic dimension $\mathrm{non}(\mathcal M)$, which denotes the smallest cardinality of a non-meager subset of $\mathbb R$. This strengthens a theorem of Sobota. More generally, we prove that every Banach space with density character $κ$ contains a barrelled subspace with algebraic dimension $\mathrm{cf}[κ]^ω\cdot \mathrm{non}(\mathcal M)$, and in particular it is consistent with $\mathsf{ZFC}$ that every Banach space with density character $<\!\mathfrak{c}$ has a barrelled subspace with dimension $<\!\mathfrak{c}$. We also prove that if the dual of a Banach space contains either $c_0$ or $\ell^p$ for some $p \geq 1$, then that space does not have a barrelled subspace with dimension $<\!\mathrm{cov}(\mathcal N)$, which denotes the smallest cardinality of a collection of Lebesgue null sets covering $\mathbb R$. In particular, it is consistent with $\mathsf{ZFC}$ that no classical Banach spaces contain barrelled subspaces with dimension $\mathfrak{b}$. This partly answers a question of Sánchez Ruiz and Saxon. |
| title | Small-dimensional normed barrelled spaces |
| topic | Functional Analysis Logic |
| url | https://arxiv.org/abs/2410.13970 |