The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets
type: Item
instance of: scholarly article
title: The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets
zbMATH Open document ID: 1118.46014
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-14T00:00:00Z
review text: The Krein-Šmulian theorem, which states that the closed convex hull of weakly compact subset of a Banach space is weakly compact, may be stated equivalently as follows: If \(X\) is a Banach space and \(K\subset X^{**}\) is a \(w^*\)-compact subset such that the distance \(d(K,X) = 0\), then \(d(\overline{co}^{w^*}(K), X) =0\). The natural question then arises, for what Banach spaces holds \(d(K,X) = d(\overline{co}^{w^*}(K), X)\) for any \(w^*\)-compact set \(K\subset X^{**}\)? Although the equality is not satisfied for all Banach spaces, it still holds for a fairly large class of spaces. The author shows here that in the case when \(X=\ell_\varphi(I)\), equipped with either the Luxemburg or Orlicz norm, equality holds if and only if \(\varphi\) satisfies the \(\Delta_2\)-condition. The second main result states that if \(C\subset X\) is nonempty and convex, and \(K\subset X^{**}\) is \(w^*\)-compact, then \(d(\overline{co}^{w^*}(K),C) \leq 9d(K,C)\); and if \(K\cap C\) is \(w^*\)-dense in \(K\), then \(d(\overline{co}^{w^*}(K), C) \leq 4 d(K,C)\). This is a generalization of the author's previous results stated for \(C=X\) [Rev. Mat. Iberoam. 22, No. 1, 93--110 (2006; Zbl 1117.46002)].
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 5125980
zbMATH Keywords: Krein-Šmulian theorem
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: convex sets
reviewed by: Anna Kamińska
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.03.090
OpenAlex ID: W2072596556
cites work: Q4338822
cites work: Q4148368
cites work: A quantitative version of Krein's theorem
cites work: An extension of the Krein--Smulian theorem
cites work: The classical Banach spaces ℓ_{𝑣𝑎𝑟𝑝ℎ𝑖}/ℎ_{𝜑}
cites work: Convexity and \(w^*\)-compactness in Banach spaces
cites work: Q3272715
cites work: Subspaces of certain Banach Sequence Spaces
DOI: 10.1016/J.JMAA.2006.03.090
Recommended article: An extension of the Krein--Smulian theorem
Recommended article: Q5324220
Recommended article: An Eberlein-Šmulian type result for the weak* topology
Recommended article: A quantitative version of Krein's theorem
Recommended article: Convexity and \(w^*\)-compactness in Banach spaces
Recommended article: The class of universally Krein-Šmulian Banach spaces
Recommended article: Farthest points in W*-compact sets
Recommended article: Q5498346
Recommended article: New results on weakly compact sets
Recommended article: Q3170652
description: scientific article; zbMATH DE number 5125980