On nested sequences of convex sets in Banach spaces
type: Item
instance of: scholarly article
title: On nested sequences of convex sets in Banach spaces
zbMATH Open document ID: 1241.46007
author: Nilson C. jun. Bernardes
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-02-27T00:00:00Z
review text: Let \(X\) be a real or complex Banach space and \((C_n)\) be a nested sequence of nonempty closed, bounded convex subsets of \(X\). By a surprisingly elementary argument it is established that (Theorem 1) \[ \forall x\in X,\;\lim_{n\to\infty} d(x,C_n)=d\left(x,\bigcap_{n}\overline{C_n}^{w^\ast}\right). \] This result was previously proved for separable spaces by \textit{J. M. F. Castillo} and \textit{P. L. Papini} [``Approximation of the limit distance function in Banach spaces'', J. Math. Anal. Appl. 328, No. 1, 577--589 (2007; Zbl 1119.46017)], where it is also shown that convexity is essential. If \((C_n)\) is contained in some weakly compact set \(C\) of \(X\), then of course the above result has to be true without any \(w^\ast\)-closure. Indeed (Theorem 2), \(C\) is weakly compact if and only if the above theorem holds without \(w^\ast\)-closure for all choices of \((C_n)\) inside \(C\). Here James's compactness theorem plays a key role for the proof. The remaining part concerns two related questions on approximating \(w^*\)-compact, convex sets in \(X^{\ast\ast}\) when \(X^\ast\) is separable, posed by Castillo and Papini. The author obtains (Theorem 3) that when \(X^\ast\) is separable, then for every \(w^*\)-compact, convex set \(C\) in \(X^{\ast\ast}\), there exists a nested sequence \((C_n)\) of closed, bounded and convex sets in \(X\) such that \(C=\bigcap_n \overline{C_n}^{w^\ast}\).
reviewed by: Olav Nygaard
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 6009681
zbMATH Keywords: nested sequence of sets
zbMATH Keywords: distance function
zbMATH Keywords: weak compactness
zbMATH Keywords: convex sets
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.12.002
OpenAlex ID: W2002182564
cites work: Q3744860
cites work: Convergence of decreasing sequences of convex sets in nonreflexive Banach spaces
cites work: Distance types in Banach spaces
cites work: Approximation of the limit distance function in Banach spaces
cites work: Application des ultraproduits à l'étude des espaces et des algèbres de Banach
cites work: Ultraproducts in Banach space theory.
cites work: Weakly Compact Sets
cites work: Convex Bodies and Periodic Homeomorphisms in Hilbert Space
cites work: A Note on Topological Properties of Normed Linear Spaces
cites work: The geometry of nested families with empty intersection. structure of the unit sphere of a nonreflexive space
DOI: 10.1016/J.JMAA.2011.12.002
Recommended article: Farthest points in W*-compact sets
Recommended article: New results on weakly compact sets
Recommended article: Q3170652
Recommended article: An example of a bounded approximatively compact set which is not locally compact
Recommended article: Krein-Milman's extreme point theorem and weak topology on hyperspace
Recommended article: Q5498346
Recommended article: Q5026401
Recommended article: Characterization of approximative compactness in Banach spaces
Recommended article: Convexity and \(w^*\)-compactness in Banach spaces
Recommended article: Characterizations of weakly compact sets and new fixed point free maps in c0
description: scientific article; zbMATH DE number 6009681