Survey on Baire-type properties in metrizablec0(Omega, X)
[EN] In [1] it was proved that if $\Omega $ is a non-empty set and $X$ is a normed space, the normed space $c_{0}(\Omega ,X)$ is barrelled, ultrabornological or unordered Baire-like if and only if $X$ is, respectively, barrelled, ultrabornological or unordered Baire-like. If $X$ is a metrizable loca...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Politècnica de València (UPV) |
| Repositorio: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglés |
| OAI Identifier: | oai:dnet:riunet______::dedc1f2a7873b8666f2c9055e7b4f477 |
| Acceso en línea: | https://riunet.upv.es/handle/10251/236104 |
| Access Level: | acceso abierto |
| Palabra clave: | Banach disk Baire Baire-like Baireled Metrizable Unordered Baire-like |
| Sumario: | [EN] In [1] it was proved that if $\Omega $ is a non-empty set and $X$ is a normed space, the normed space $c_{0}(\Omega ,X)$ is barrelled, ultrabornological or unordered Baire-like if and only if $X$ is, respectively, barrelled, ultrabornological or unordered Baire-like. If $X$ is a metrizable locally convex space, and $\{\left\Vert \cdot \right\Vert_{n}:n\in \mathbb{N\}}$ is an increasing sequence of semi-norms defining its topology, $c_{0}(\Omega ,X)$ is the metrizable locally convex space over the field $\mathbb{K}$ (of the real or complex numbers) of all functions $f:\Omega \rightarrow X$ such that for each $\epsilon >0$ and $n\in \mathbb{N}$ the set $\left\{ \omega \in \Omega :\left\Vert f(\omega )\right\Vert_{n}>\epsilon \right\} $ is finite or empty, with the topology defined by the semi-norms $\left\Vert f\right\Vert _{n}:=\sup \left\{ \left\Vertf(\omega )\right\Vert _{n}:\omega \in \Omega \right\} $, $n\in \mathbb{N}$. Also, in [2], it was proved that the metrizable $c_{0}(\Omega,X)$ is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class $p$ if and only if $X$ is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class $p$. In [3] it was proved that the metrizable $c_{0}(\Omega ,X)$ is baireled if and only if $X$ is baireled. Two open problems are presented. \bigskip [1] J. C. Ferrando and S. V. L\"{u}dkovsky, Some barrelledness properties of $c_{0}(\Omega ,X)$, J. Math. Anal. Appl. 274, 577--585, 2002. [2] J. K\c{a}kol, M. L\'{o}pez Pellicer and S. Moll-L\'{o}pez, Banach disks and barrelledness properties of metrizable c0(\U{3a9}, X), Mediterr. J. Math. 1, 81--91, 2004. [3] S. L\'{o}pez-Alfonso, M. L\'{o}pez-Pellicer and S. Moll-L\'{o}pez, Baire-type properties in metrizable c0(\U{3a9}, X), Axioms 11 (1), 2022; Article ID: 6. |
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