Limited type subsets of locally convex spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911788635783168 |
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| author | Gabriyelyan, Saak |
| author_facet | Gabriyelyan, Saak |
| contents | Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse $p$-limited subsets of an arbitrary locally convex space $E$. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact and relatively weakly sequentially (pre)compact sets. If $E$ is a Banach space, we show that the class of coarse $1$-limited subsets of $E$ coincides with the class of $(1,\infty)$-limited sets, and if $1<p<\infty$, then the class of coarse $p$-limited sets in $E$ coincides with the class of $p$-$(V^\ast)$ sets of Pełczyński. We also generalize a known theorem of Grothendieck. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02016 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limited type subsets of locally convex spaces Gabriyelyan, Saak Functional Analysis 46A3, 46E10 Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse $p$-limited subsets of an arbitrary locally convex space $E$. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact and relatively weakly sequentially (pre)compact sets. If $E$ is a Banach space, we show that the class of coarse $1$-limited subsets of $E$ coincides with the class of $(1,\infty)$-limited sets, and if $1<p<\infty$, then the class of coarse $p$-limited sets in $E$ coincides with the class of $p$-$(V^\ast)$ sets of Pełczyński. We also generalize a known theorem of Grothendieck. |
| title | Limited type subsets of locally convex spaces |
| topic | Functional Analysis 46A3, 46E10 |
| url | https://arxiv.org/abs/2403.02016 |