Characterization of barrelled, d-barrelled, and m\(\sigma\)-barrelled spaces of continuous functions
type: Item
instance of: scholarly article
title: Characterization of barrelled, d-barrelled, and m\(\sigma\)-barrelled spaces of continuous functions
zbMATH Open document ID: 0585.46020
DOI: 10.1007/BF01157977
author: I. I. Perepechaj
published in: Mathematical Notes
publication date: 1985-01-01T00:00:00Z
review text: Let T be a completely regular Hausdorff space and C(T) the space of all continuous real-valued functions on T. Let \({\mathcal P}\) be a family of subsets of T on each of which every \(f\in C(T)\) is bounded. Let \(C_{{\mathcal P}}(T)\) be C(T) provided with the topology of uniform convergence on members of \({\mathcal P}\). The barrelled, d-barrelled and \(\sigma\)-barrelled spaces of type \(C_{{\mathcal P}}(T)\) were characterized in the book.'' Espaces de Fonctions continues'' of \textit{J. Schmets} (1976; Zbl 0334.46022). The present author makes a similar study of \(C_{{\mathcal P}}(T)\) viewed as a topological vector lattice in the sense of \textit{J. Jameson} [Math. Z. 103, 139-150 (1968; Zbl 0173.153)] where this kind of structure is called an M-space. He defines the m-barrelled, md-barrelled and \(m\sigma\)-barrelled M-spaces and characterizes the spaces \(C_{{\mathcal P}}(T)\) of this type. It turns out that \(C_{{\mathcal P}}(T)\) is m-barrelled (resp. md-barrelled) if and only if it is barrelled (resp. d-barrelled). The author notes that it is unknown whether the analogous result holds for the \(\sigma\)- barrelledness property.
Mathematics Subject Classification ID: 46E05
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46A08
zbMATH DE Number: 3937802
zbMATH Keywords: vector lattice
zbMATH Keywords: space of continuous function
zbMATH Keywords: M-space
zbMATH Keywords: m-barrelled
zbMATH Keywords: md- barrelled
zbMATH Keywords: m\(\sigma \) -barrelled
reviewed by: Willy J. F. Govaerts
MaRDI profile type: Publication
cites work: Q4057135
cites work: Topological M-spaces
full work available at URL: https://doi.org/10.1007/bf01157977
OpenAlex ID: W2050892981
description: scientific article; zbMATH DE number 3937802