Some geometric and topological properties of Banach spaces via ball coverings
type: Item
instance of: scholarly article
title: Some geometric and topological properties of Banach spaces via ball coverings
zbMATH Open document ID: 1218.46009
author: Bo Wang
author: Wen Zhang
author: Yu Zhou
author: Lixin Cheng
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-03-08T00:00:00Z
review text: The \textit{ball topology} on a Banach space \(X\) was studied by \textit{G. Godefroy} and \textit{N. J. Kalton} [Banach space theory, Proc. Res. Workshop, Iowa City/Iowa 1987, Contemp. Math. 85, 195--237 (1989; Zbl 0676.46003)]. By definition it is the coarsest topology \(b_ X\) such that every closed ball of \(X\) is a closed set in \(b_ X\). \(X\) is said to have the \textit{ball-covering property} if the unit sphere of \(X\) can be covered by a countable collection of balls that do not contain the origin. It is shown that \(X\) has the ball-covering property if and only if every point of \(X\) is a \(G_\delta\) set in the ball topology. Some characterizations of smoothness and uniform smoothness of \(X\) in therms of ball-coverings of finite dimensional subspaces are given.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5862738
zbMATH Keywords: ball-covering
zbMATH Keywords: ball topology
zbMATH Keywords: smoothness
zbMATH Keywords: Banach space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.12.013
OpenAlex ID: W2031097272
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DOI: 10.1016/J.JMAA.2010.12.013
description: scientific article; zbMATH DE number 5862738