Mosco convergence in locally convex spaces
type: Item
instance of: scholarly article
title: Mosco convergence in locally convex spaces
zbMATH Open document ID: 0781.46020
DOI: 10.1016/0022-1236(92)90047-M
published in: Journal of Functional Analysis
publication date: 1993-06-29T00:00:00Z
review text: Given a dual pair \(E\), \(F\) of locally convex spaces, each with its corresponding weak topology \(\sigma\) and Mackey topology \(\tau\), one says that a sequence \(\{f_ n\}\) of functions \(E\to [-\infty,\infty]\) (or \(F\to [-\infty,\infty]\)) is Mosco-convergent to a function \(f_ 0\) if the following conditions are satisfied for each \(v\) in \(E\) (or in \(F\)): (i) \(\liminf f_ n(v_ n)\geq f_ 0(v)\) for every sequence \(\{v_ n\}\) \(\sigma\)-convergent to \(v\), (ii) \(\limsup f_ n(v_ n)\leq f_ 0(v)\) for some sequence \(\{v_ n\}\) \(\tau\)-convergent to \(v\). Mosco showed that if \(E\) is a reflexive Banach space, and \(F\) is its dual, and if each \(f_ n\) is a proper lower-semicontinuous convex function, then \(\{f_ n\}\) is Mosco-convergent to \(f_ 0\) if and only if \(\{f_ n^*\}\) is Mosco-convergent to \(f^*_ 0\), the convex conjugate of \(f_ 0\). After some relevant exposition, the second of these implications is extended to the case in which \(F\) is a separable Fréchet space dual to a locally convex space \(E\). Some related results are discussed.
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 52A41
zbMATH DE Number: 220682
zbMATH Keywords: weak topology
zbMATH Keywords: Mackey topology
zbMATH Keywords: Mosco-convergent
zbMATH Keywords: reflexive Banach space
zbMATH Keywords: proper lower-semicontinuous convex function
zbMATH Keywords: convex conjugate
zbMATH Keywords: separable Fréchet space dual
author: Sandy L. Zabell
reviewed by: J. D. Weston
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 220682