On the Dieudonné theorem in reflexive Banach spaces
type: Item
instance of: scholarly article
title: On the Dieudonné theorem in reflexive Banach spaces
zbMATH Open document ID: 1054.46012
published in: Cybernetics and Systems Analysis
publication date: 2003-09-09T00:00:00Z
review text: The authors first characterize local compactness of closed convex sets containing no lines in a reflexive Banach space provided with its weak topology by a geometrical condition, which they call well-positionedness. For the proof of this characterization they utilize technical results of a forthcoming article of their own, which has not yet appeared. Next, they investigate the Dieudonné theorem about the closedness of the algebraic difference of two closed convex sets and prove the following Theorem: A closed convex subset \(C_1\) containing no lines in a reflexive Banach space \(X\) is well-positioned if and only if, for every closed convex subset \(C_2\) in \(X\) such that \(C^\infty_1\cap C^\infty_2= \{0\}\), the algebraic difference \(C_1- C_2\) is closed (where \(C^\infty:= \{d\in X:\{c+\mu d: c\in C,\,\mu> 0\}\subset C\}\)). From these two results, they obtain that Dieudonné's sufficient condition becomes also a necessary one -- at least in reflexive Banach spaces with their weak topology.
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46A55
zbMATH DE Number: 1978188
zbMATH Keywords: Dieudonné theorem
zbMATH Keywords: reflexive Banach spaces
zbMATH Keywords: weak topology
author: Samir Adly
author: E. Ernst
author: Michel Théra
reviewed by: Susanne Dierolf
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1978188