Barrelledness and bornological conditions on spaces of vector-valued \(\mu\)-simple functions
type: Item
instance of: scholarly article
title: Barrelledness and bornological conditions on spaces of vector-valued \(\mu\)-simple functions
zbMATH Open document ID: 0813.46001
DOI: 10.1007/BF03323086
author: Santiago Díaz
author: Antonio Fernández
author: Miguel Florencio
author: Pedro J. Paúl
author: Lech Drewnowski
published in: Results in Mathematics
publication date: 1992-09-27T00:00:00Z
review text: The space \(S(\mu, E)\) of classes of \(\mu\)-a.e.-equal simple functions defined on a non-trivial measure space with values in a locally convex space \(E\) is considered with the topology of uniform convergence. It is shown that \(S(\mu, E)\) is quasi-barrelled (resp. bornological) if and only if \(E\) is of the same type and \(E_ \beta^ \prime \) has property \((B)\) of Pietsch. Furthermore \(S(\mu, E)\) is barrelled (resp. DF) if and only if \(E\) is barrelled and nuclear (resp. \(E\) is DF). The space \(S(\mu, E)\) is never ultrabornological. The proofs are based on the isomorphism \(E(\mu, E)\cong S(\mu, \mathbb{K})\otimes_ \varepsilon E\) and on Stone's representation theorem.
reviewed by: Andreas Kriegl
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 28B05
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46A32
zbMATH DE Number: 62169
zbMATH Keywords: barrelledness conditions
zbMATH Keywords: injective tensor product
zbMATH Keywords: nuclearity
zbMATH Keywords: simple functions
zbMATH Keywords: quasi-barrelled
zbMATH Keywords: bornological
zbMATH Keywords: barrelled and nuclear
zbMATH Keywords: ultrabornological
zbMATH Keywords: Stone's representation theorem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03323086
OpenAlex ID: W2049571683
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description: scientific article; zbMATH DE number 62169