Relations between certain classes of sets in Banach spaces
type: Item
instance of: scholarly article
title: Relations between certain classes of sets in Banach spaces
zbMATH Open document ID: 0625.46017
DOI: 10.1007/BF01159114
author: Igor G. Tsar'kov
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: Let X be a real normed linear space and M a non-empty subset of X. For \(x\in X\) write \(P_ Mx=\{y\in M:\| x-y\| =d(x,M)\}\), where \(d(x,M)=\inf \{\| x-m\|:m\in M\}\). M is called approximately compact if for each sequence \(\{y_ n\}\) in M such that \(\| x-y_ n\| \to d(x,M)\) there exists a subsequence converging in M. Denote by \({\mathcal Z}\) the set of all non-empty closed subsets of X and by \((P_ 0)\) (respectively (P)) the family of subsets M of X for which \(P_ Mx\) is connected (respectively connected and non-empty) for all x. The following results are typical for the theorems proved in the paper: (1) Let X be a Banach space with the property that each hyperplane is approximately compact. Then each \(M\in (P_ 0)\cap {\mathcal Z}\) has the property that \(M\cap \{y:\| y-x\| <r\}\) is connected for all x and all \(r>0.\) (2) \((P_ 0)\cap {\mathcal Z}=(P)\) if and only if X is finite-dimensional.
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 4015435
zbMATH Keywords: Efimov-Stechkin spaces
zbMATH Keywords: Chebyshev sets
zbMATH Keywords: approximately compact
zbMATH Keywords: hyperplane
MaRDI profile type: Publication
cites work: Q3286569
cites work: Characterizations of reflexivity
cites work: APPROXIMATIVE PROPERTIES OF SETS IN NORMED LINEAR SPACES
cites work: Chebyshev Sets in C[0,1] Which are not Suns
cites work: A Note on the Pseudo-Arc
cites work: Q4183176
cites work: Q3213662
cites work: Q3267713
cites work: Q5536924
full work available at URL: https://doi.org/10.1007/bf01159114
OpenAlex ID: W1980080654
description: scientific article; zbMATH DE number 4015435