The domination property for efficiency and Bishop-Phelps theorem in locally convex spaces
type: Item
instance of: scholarly article
title: The domination property for efficiency and Bishop-Phelps theorem in locally convex spaces
zbMATH Open document ID: 1273.46001
author: Jinghui Qiu
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-04-22T00:00:00Z
review text: Let \(X\) be a locally convex space (LCS) and \(S\) a cone in \(X\). A point \(x\) in a nonempty subset \(A\) of \(X\) is called an efficient point for \(A\) with respect to \(S\) if \((A-x)\cap (-S)=\{0\}\). The set of efficient points is denoted by \(E(S,A)\). One says that \(A\) has the domination property with respect to \(S\) if every point in \(A\) is dominated by an efficient point. For an absolutely convex subset \(B\) of \(X\), one denotes by \(E_B\) the space spanned by \(B\) equipped with the norm \(p_B\), where \(p_B\) is the Minkowski functional on \(E_B\). One says that \(B\) is a Banach disc if \((E_B,p_B)\) is a Banach space. One defines in terms of these normed subspaces localized versions of topological notions: local convergence of sequences in \(X\), local completeness of \(X\), local closedness of subsets, etc. (see, for instance, [\textit{P. Pérez Carreras} and \textit{J. Bonet}, Barrelled locally convex spaces. Amsterdam etc.: North-Holland (1987; Zbl 0614.46001)]). For a convex subset \(\Theta\) of \(X\), one also defines the notions of \(\Theta\)-closure and \(\Theta\)-closedeness, weaker than the corresponding usual notions. A \(\sigma\)-convex set (or CS-compact) is a subset \(C\) of \(X\) such that every convex series in \(C\) converges to an element of \(C\). \textit{D. H. Fremlin} and \textit{M. Talagrand} [Mathematika 26, 30--32 (1979; Zbl 0456.46011)] proved that every bounded convex \(G_\delta\)-subset of a complete LCS is \(\sigma\)-convex. In Proposition 4.2, this result is extended to locally complete LCS. The main result on the domination property (Theorem 5.1) asserts that, if \(S\) is a convex cone with a \(\sigma\)-convex base in a LCS \(X\), then every nonempty subset \(A\) of \(X\) for which there exists a linear functional \(f\) that is lower bounded on \(A\) has the domination property with respect to \(A\) (in particular, \(E(S,A)\neq\emptyset\)). Defining a notion of \(B\)-support point of a subset \(A\) of \(X\), where …
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 6156108
zbMATH Keywords: locally convex spaces
zbMATH Keywords: ordered locally convex space
zbMATH Keywords: cone
zbMATH Keywords: cone base
zbMATH Keywords: efficient points
zbMATH Keywords: Banach disc
zbMATH Keywords: Bishop-Phelps property
zbMATH Keywords: convex series property
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.12.072
OpenAlex ID: W2086596755
DOI: 10.1016/J.JMAA.2012.12.072
description: scientific article; zbMATH DE number 6156108