Barrelled and bornological function spaces
type: Item
instance of: scholarly article
title: Barrelled and bornological function spaces
zbMATH Open document ID: 0969.46003
author: Julian Dontchev
author: Sergio Salbany
author: V. Valov
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-10-02T00:00:00Z
review text: A subset \(K\) of a completely regular topological space \(Y\) is \(t\)-bounded if \(f(K)\) is a bounded subset of \(\mathbb{R}\) for every real continuous function \(f\) on \(Y\). Let \(X\) be a completely regular topological space. \(\nu X\) is the real-compactification of \(X\). Let \(\lambda\) be a cover of \(X\) such that every element of \(\lambda\) is closed and \(t\)-bounded in \(X\) and \(\lambda\) is invariant with respect to finite unions. \(\nu_\lambda(X):= \bigcup\{\text{cl}_{\nu X}(K): K\in\lambda\}\) with the topology inherited from \(\nu X\). If \(Z\) is a subspace of \(\nu X\) containing \(\nu_\lambda(X)\), then a set \(B\subset Z\) is \(\lambda\) bounded in \(Z\) if every lower semi-continuous function \(\varphi: Z\to [0,\infty)\) is bounded on \(B\) provided it is bounded on each \(K\in\lambda\). If \(E\) is a normed space, \(C_\lambda(X, E)\) denotes the set of all \(E\)-valued continuous maps on \(X\) with the topology of uniform convergence on members of \(\lambda\). The authors give the following results: 1. Let \(E\) be a Banach space. Then \(C_\lambda(X, E)\) is barrelled if and only if every \(t\)-bounded subset of \(\nu_\lambda(X)\) is contained in \(\text{cl}_{\nu X}(K)\) for some \(K\in\lambda\). 2. \(C_\lambda(X, E)\) is quasi-barrelled if and only if every \(\lambda\)-bounded subset of \(\nu_\lambda(X)\) is contained in \(\text{cl}_{\nu X}(K)\) for some \(K\in\lambda\). 3. \(C_\lambda(X, E)\) is bornological if and only if every \(\lambda\)-bounded subset of \(\nu X\) is contained in \(\text{cl}_{\nu X}(K)\) for some \(K\in\lambda\). 4. Let \(E\) be a Banach space. Then \(C_\lambda(X, E)\) is ultrabornological if and only if every \(t\)-bounded subset of \(\nu X\) is contained in \(\text{cl}_{\nu K}(K)\) for some \(K\in\lambda\).
reviewed by: Manuel Valdivia
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 1423635
zbMATH Keywords: completely regular topological space
zbMATH Keywords: \(t\)-bounded
zbMATH Keywords: real-compactification
zbMATH Keywords: barrelled
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6625
OpenAlex ID: W2029549428
cites work: Q3672683
cites work: Q4228099
cites work: Limitation Topologies on Function Spaces
cites work: Q4000501
cites work: Q4118337
cites work: Q4150804
cites work: Closure-preserving covers by small sets
cites work: Q4080169
cites work: Q3310575
cites work: Function spaces
DOI: 10.1006/JMAA.1999.6625
Recommended article: Q4015693
Recommended article: Barrelledness and bornological conditions on spaces of vector-valued \(\mu\)-simple functions
Recommended article: On the barrelledness of spaces of bounded vector functions
Recommended article: Q3153082
Recommended article: On the barrelledness of the vector-valued bounded function space
Recommended article: Q4017102
Recommended article: Bornologies and bitopological function spaces
Recommended article: Barrelledness of p-Adic C1-Function Spaces
Recommended article: Characterization of barrelled, d-barrelled, and m\(\sigma\)-barrelled spaces of continuous functions
Recommended article: On \(s\)-barrelled spaces
description: scientific article; zbMATH DE number 1423635