Weak normal structure in Banach spaces with symmetric norm
type: Item
instance of: scholarly article
title: Weak normal structure in Banach spaces with symmetric norm
zbMATH Open document ID: 0952.46013
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-01-15T00:00:00Z
review text: Let \(X\) be a Banach space with weak normal structure. Then, \(X\) has the fixed-point property; that is, every non-expansive map from a weakly compact subset into itself has a fixed point, which was proved by \textit{W. A. Kirk} [Am. Math. Monthly 72, 1004-1006 (1995; Zbl 0141.32402)]. In this note, the authors claim the following theorem: Let \(X\) be a separable Banach space. Then \(X\) can be renormed to fail to have weak normal structure if and only if \(X\) is not a Schur space. A Schur space is a Banach space in which a sequence converges weakly if and only if it converges in norm.
reviewed by: Shozo Koshi
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 1340099
zbMATH Keywords: Banach space with weak normal structure
zbMATH Keywords: fixed-point property
zbMATH Keywords: non-expansive map
zbMATH Keywords: renormed
zbMATH Keywords: Schur space
author: Helga Fetter
author: Berta Gamboa de Buen
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6419
OpenAlex ID: W2053710496
cites work: Q3346801
cites work: Q5786938
cites work: Normed Linear Spaces that are Uniformly Convex in Every Direction
cites work: A Fixed Point Theorem for Mappings which do not Increase Distances
cites work: Q4136031
cites work: Normal structure in Banach spaces and its generalisations
DOI: 10.1006/JMAA.1999.6419
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description: scientific article; zbMATH DE number 1340099