Local completeness and drop theorem
type: Item
instance of: scholarly article
title: Local completeness and drop theorem
zbMATH Open document ID: 1004.46012
author: Jinghui Qiu
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-01-30T00:00:00Z
review text: Let \(E\) be a Banach space with closed unit ball \(B\) and \(x\in E\setminus B\). The ``drop'' generated by \(x\) and \(B\) is the set \(D(x,B)=\operatorname {co}(B\cup\{x\})\). Danes drop theorem asserts that for every nonempty closed set \(A\subset E\) situated at a positive distance from \(B\) there exists \(a\in A\) such that \(D(a,B)\cap A = \{a\}\) [\textit{J. Daneš}, Boll. Un. Mat. Ital. 6, 369--372 (1972; Zbl 0236.47053) and Commentat. Math. Univ. Carol. 26, 443--454 (1985; Zbl 0656.47050)]. The drop property turned to be a very important and useful geometric tool, equivalent to Ekeland's variational principle [see \textit{J.-P. Penot}, Nonlinear Analysis 10, 813--822 (1986; Zbl 0612.49011)]. The aim of the present paper is to extend this property to locally convex spaces. The main innovation of the author consists in the use of the notions of local closedness and local completeness of subsets of a locally convex space \(E\), defined in terms of the Minkowski gauge of some discs in \(E\). These notions are weaker than those of sequential closedness and sequential completeness, used by \textit{L. Cheng, Y. Zhou} and \textit{F. Zhang} [Proc. Am. Math. Soc 124, No. 12, 3699--3702 (1996; Zbl 0863.46003)], to prove a drop theorem in locally convex spaces.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46A50
zbMATH DE Number: 1744276
zbMATH Keywords: drop property
zbMATH Keywords: Danes' drop theorem
zbMATH Keywords: Ekeland's variational principle
zbMATH Keywords: drop theorem in locally convex spaces
zbMATH Keywords: local closedness
zbMATH Keywords: local completeness
zbMATH Keywords: Minkowski gauge
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: jmaa.2001.7709
OpenAlex ID: W2026074547
cites work: Q5645986
cites work: Q3803705
cites work: A general principle on ordered sets in nonlinear functional analysis
cites work: Normal solvability and the Fredholm alternative for mappings into infinite dimensional manifolds
cites work: Q4741204
cites work: The drop theorem, the petal theorem and Ekeland's variational principle
cites work: Q3784621
cites work: On the drop and weak drop properties for a Banach space
cites work: Characterisation of drop and weak drop properties for closed bounded convex sets
cites work: Danes’ Drop Theorem in locally convex spaces
cites work: Q4486063
cites work: Q5518756
cites work: Q4722576
cites work: Locally convex spaces containing no copy of \(c_ 0\)
DOI: 10.1006/JMAA.2001.7709
description: scientific article; zbMATH DE number 1744276