Every Banach space with a \(w^{*}\)-separable dual has a \(1+\varepsilon\)-equivalent norm with the ball covering property
type: Item
instance of: scholarly article
title: Every Banach space with a \(w^{*}\)-separable dual has a \(1+\varepsilon\)-equivalent norm with the ball covering property
zbMATH Open document ID: 1191.46010
author: Wen Zhang
author: Lixin Cheng
author: Huihua Shi
published in: Science in China. Series A
publication date: 2009-12-11T00:00:00Z
review text: A \textit{countable ball-covering} in a Banach space~\(X\) a is countable collection of balls off the origin such that the union of these balls contains the unit sphere of~\(X\). This concept was introduced by \textit{L.-X.\thinspace Cheng} in [Isr.\ J.\ Math.\ 156, 111--123 (2006; Zbl 1139.46016)]. It easily follows from the Hahn-Banach separation theorem that, if \(X\) admits a countable ball-covering, then \(X^*\) is weak\(^*\)-separable. The authors prove a converse theorem under renorming: if \(X^*\) is weak\(^*\)-separable, then for every \(\varepsilon > 0\) the space \(X\) possesses a \((1+\varepsilon)\)-equivalent norm in which \(X\) admits a countable ball-covering. The same result was obtained independently in [\textit{V.\,P.\thinspace Fonf} and \textit{C.\,Zanco}, Math.\ Ann.\ 344, No.\,4, 939--945 (2009; Zbl 1179.46015)].
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B26
zbMATH DE Number: 5645734
zbMATH Keywords: ball-covering
zbMATH Keywords: Banach space
zbMATH Keywords: equivalent norm
MaRDI profile type: Publication
full work available at URL: s11425-009-0175-7
OpenAlex ID: W2039710329
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DOI: 10.1007/S11425-009-0175-7
description: scientific article; zbMATH DE number 5645734