On compactness in locally convex spaces
type: Item
instance of: scholarly article
title: On compactness in locally convex spaces
zbMATH Open document ID: 0604.46011
DOI: 10.1007/BF01161762
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183698
review text: We describe a class \({\mathcal G}\) of locally convex spaces that contains the metrizable spaces and the dual metric spaces and such that: (1) \({\mathcal G}\) is stable by the following operations: arbitrary subspaces, separated quotients, countable locally convex direct sums, countable products and completions. (2) Every space of the class \({\mathcal G}\) has metrizable precompact subsets. (3) Every space of the class \({\mathcal G}\) is weakly angelic. (4) p4 A weakly compact subset in a space of the class \({\mathcal G}\) is metrizable if, and only if, it is contained in a separable subspace. (5) A compact topological space K is such that the Banach space \(C(K)\) is weakly \(K\)-analytic if, and only if, \(K\) is homeomorphic to a weakly compact subset of a locally convex space of the class \({\mathcal G}.\) Applications to spaces of vector-valued continuous functions, to locally convex spaces with analytic duals and to the general problem of retractivity in inductive limits are also given.
Mathematics Subject Classification ID: 46A50
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 3975618
zbMATH Keywords: metrizable spaces
zbMATH Keywords: dual metric spaces
zbMATH Keywords: metrizable precompact subsets
zbMATH Keywords: weakly angelic
zbMATH Keywords: K-analytic
zbMATH Keywords: spaces of vector-valued continuous functions
zbMATH Keywords: locally convex spaces with analytic duals
zbMATH Keywords: retractivity in inductive limits
author: B. Cascales
author: José Orihuela
MaRDI profile type: Publication
cites work: The structure of weakly compact sets in Banach spaces
cites work: Q4741006
cites work: Q3811158
cites work: On K-analytic locally convex spaces
cites work: Metrizability of precompact subsets in (LF)-spaces
cites work: Pointwise compactness in spaces of functions and R. C. James theorem
cites work: Q4150677
cites work: Weak Compactness in Banach Spaces
cites work: Q4150804
cites work: Folgenretraktive Sequenzen lokalkonvexer Räume.
cites work: Weakly compact sets. Lectures held at S.U.N.Y., Buffalo, in spring 1978
cites work: Criteres de Compacite dans les Espaces Fonctionnels Generaux
cites work: Q5832620
cites work: Q3918541
cites work: Q5569793
cites work: Über die Regularitätsbegriffe induktiver lokalkonvexer Sequenzen
cites work: Pointwise Compactness in Spaces of Continuous Functions
cites work: Summable sequences and topological properties of \(m_0(I)\)
cites work: On Weakly Compact Subsets of a Banach Space
cites work: A Device Of R. J. Whitley's Applied to Pointwise Compactness in Spaces of Continuous Functions
cites work: K ‐analytic sets
cites work: Espaces de Banachs faiblement K-analytiques
cites work: Q3951046
cites work: Quasi-LB-Spaces
cites work: A survey of Baire measures and strict topologies
description: scientific article; zbMATH DE number 3975618