Weakly compactly LUR Banach spaces
type: Item
instance of: scholarly article
title: Weakly compactly LUR Banach spaces
zbMATH Open document ID: 1391.46023
author: P. Shunmugaraj
author: S. Dutta
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-11-28T00:00:00Z
review text: Variants of the well-known concept of local uniform rotundity (LUR, in short) and of its weak version WLUR (use weak convergence instead of norm convergence in the definition) are considered in the literature on the geometry of the closed unit ball \(B_X\) of a Banach space \(X\). The space \(X\) is said to be compactly locally uniformly rotund (CLUR, in short) if \(\{x_n\}\) has a norm-convergent subsequence whenever \(x, x_n\in S_X\) and \(\|x+x_n\|\rightarrow 2\), where \(S_X\) denotes the unit sphere of \(X\). If the convergence of the subsequence is in the weak topology, the space is said to be WCLR (a concept due to L. P. Vlasov). In the present paper, the class of weakly CLUR is introduced: As above, the sequence \(\{x_n\}\) has a subsequence that weakly converges, but now the weak limit is in \(A_0(x):=\bigcup_{x^*\in J(x)} S(X,x^*,0)\), where \(J(x):=\{x^*\in S_{X^*}:\;x^*(x)=1\}\), and \(S(X,x^*,\delta):=\{y\in B_{X}:\;x^*(y)\geq 1-\delta\}\) for \(\delta\geq 0\). Two examples are presented to help separating some of those concepts: Both are renormings of \(\ell_2\). The first one presents a (reflexive) space with the Kadets-Klee property (KK, in short, i.e., on \(S_X\) the weak and norm-convergent sequences coincide) that is not weakly CLUR. The second one is (reflexive and) weakly CLUR, although it is neither CLUR nor WLUR. The remainder of the paper deals with characterizations of the weakly CLUR and the CLUR properties, gives a sufficient condition for weakly CLUR on reflexive spaces in terms of the KK property, and, alternatively, in terms of the local nearly uniformly smooth property or in terms of the strong subdifferentiability of the norm together with the compactness of the duality mapping (two sets of conditions that, according to a later proposition, turn out to be the same).
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 6813447
zbMATH Keywords: weakly compactly locally uniformly convex
zbMATH Keywords: locally nearly uniformly convex
zbMATH Keywords: locally nearly uniformly smooth
zbMATH Keywords: Kadec-Klee property
zbMATH Keywords: approximatively weakly compact
reviewed by: V. Montesinos
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.09.048
OpenAlex ID: W2762279193
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DOI: 10.1016/J.JMAA.2017.09.048
description: scientific article; zbMATH DE number 6813447