Baire-type properties of topological vector spaces
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910012663660544 |
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| author | Gabriyelyan, Saak Osipov, Alexander V. Reznichenko, Evgenii |
| author_facet | Gabriyelyan, Saak Osipov, Alexander V. Reznichenko, Evgenii |
| contents | Burzyk, Kliś and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. Kcakol and Sánchez Ruiz proved that every sequentially complete Fréchet--Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $κ$-Fréchet--Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $κ$-Fréchet--Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $κ$-Fréchet--Urysohn. Although a $κ$-Fréchet--Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_23008 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Baire-type properties of topological vector spaces Gabriyelyan, Saak Osipov, Alexander V. Reznichenko, Evgenii Functional Analysis Burzyk, Kliś and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. Kcakol and Sánchez Ruiz proved that every sequentially complete Fréchet--Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $κ$-Fréchet--Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $κ$-Fréchet--Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $κ$-Fréchet--Urysohn. Although a $κ$-Fréchet--Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given. |
| title | Baire-type properties of topological vector spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2601.23008 |