Baire-Type Properties in Metrizable c(0)(omega, X)

Abstract

[EN] Ferrando and Lüdkovsky proved that for a non-empty set $\Omega $\ and a normed space $X$, 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. When $X$ is a metrizable locally convex space, with an increasing sequence of semi-norms $\left\{ \left\Vert .\right\Vert _{n}\in \mathbb{N}\right\} $ defining its topology, then $c_{0}(\Omega ,X)$ 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 $\varepsilon >0$ and $n\in\mathbb{N}$ is finite or empty, with the topology defined by the semi-norms $\left\Vert f\right\Vert _{n}=\sup \left\{ \left\Vert f(\omega )\right\Vert _{n}:\omega \in \Omega \right\} $, $n\in\mathbb{N}$. K\c{a}kol, L\'{o}pez-Pellicer and Moll-L\'{o}pez also proved that the metrizable space $c_{0}(\Omega ,X)$ is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class $p$ if and only if $X$ is, respectively, quasibarrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class $p$. The main result of this paper is that the metrizable $c_{0}(\Omega ,X)$ is baireled if and only if $X$ is baireled, and its long proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.