\(B_ r\)-complete spaces which are not B-complete
type: Item
instance of: scholarly article
title: \(B_ r\)-complete spaces which are not B-complete
zbMATH Open document ID: 0549.46005
DOI: 10.1007/BF01181696
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173399
review text: Let F be a closed subspace of \(\ell^{\infty}\). Let \(F^*\) be the closure of F in \((\ell^{\infty})''[\sigma ((\ell^{\infty})'',(\ell^{\infty})')].\) Let \({\mathcal F}_ F\) be the topology of Mackey \(\mu ((\ell^{\infty})',\ell^{\infty}+F^*).\) In this paper the following result is proved: If the strong dual of F is separable and F is of infinite codimension in \(F^*\) there is a topology \({\mathcal F}\) on \((\ell^{\infty})'\) which has the following properties: 1. \({\mathcal F}\) is coarser than \({\mathcal F}_ F\). 2. \((\ell^{\infty})'[{\mathcal F}]\) is \(B_ r\)-complete. 3. \((\ell^{\infty})'[{\mathcal F}]\) is not B-complete. A precise example of a \(B_ r\)-complete space which is not B-complete can be constructed by taking \(F=c_ 0\). An example of a \(B_ r\)-complete space with a non- complete quotient is given also.
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A20
zbMATH DE Number: 3875965
zbMATH Keywords: strong dual
zbMATH Keywords: infinite codimension
zbMATH Keywords: example of a \(B_ r\)-complete space which is not B-complete
zbMATH Keywords: example of a \(B_ r\)-complete space with a non- complete quotient
author: Manuel Valdivia
MaRDI profile type: Publication
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OpenAlex ID: W2912658871
description: scientific article; zbMATH DE number 3875965