Article ID: MTJPAM-D-23-00005

Title: Survey on Baire-type properties in metrizable c0(Ω, X)


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-23-00005; Volume 6 / Issue 3 / Year 2024, Pages 11-15

Document Type: Research Paper

Author(s): ‪Salvador López-Alfonso a , Manuel López-Pellicer b , Santiago Moll-López c

aDepartamento de Construcciones Arquitectónicas, Universitat Politècnica de València, 46022 Valencia, Spain

bIUMPA, Universitat Politècnica de València, 46022 Valencia, Spain

cDepartamento de Matemática Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain

Received: 24 March 2023, Accepted: 18 May 2023, Published: 21 June 2023

Corresponding Author: Manuel López-Pellicer (Email address: mlopezpe@mat.upv.es)

Full Text: PDF

Abstract

In [6] 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\{ \left\Vert .\right\Vert _{n}\in \mathbb{N}\right\} 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 \varepsilon >0 and n\in \mathbb{N} the set \left\{ \omega \in \Omega :\left\Vert f(\omega )\right\Vert _{n}>\varepsilon \right\} 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}. Also, in [10], 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 [14] it was proved that the metrizable c_{0}(\Omega ,X) is baireled if and only if X is baireled. Two open problems are presented.

Keywords: Banach disk, Baire, Baire-like, baireled, metrizable, unordered Baire-like

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