Pontryagin duality in the theory of topological vector spaces and in topological algebra
type: Item
instance of: scholarly article
title: Pontryagin duality in the theory of topological vector spaces and in topological algebra
zbMATH Open document ID: 1042.46002
DOI: 10.1023/A:1020929201133
published in: Journal of Mathematical Sciences (New York)
publication date: 2003-09-16T00:00:00Z
review text: This long paper presents in a unified way the recent research of the author in the theory of stereotype topological vector spaces and algebras. Let \(X\) be a Hausdorff locally convex space over \(C\), and let \(X^{*}\) denote the dual of \(X\) endowed with the topology of uniform convergence on the precompact subsets of \(X\). The author defines \(X\) to be stereotype if the canonical map \(X\to X^{**}\) is a topological isomorphism. Accordingly, a space is stereotype if and only if it is polar reflexive in the sense of \textit{G.\ Köthe} [``Topological vector spaces I'' (Springer Verlag) (1969; Zbl 0179.17001)]. The class of stereotype spaces is rather large; it contains, in particular, all quasicomplete barrelled spaces. Stereotype spaces form a category (denoted by \(\mathcal{S}\)) with a number of useful properties; in particular, it is preabelian and it is monoidal with respect to a suitably defined tensor product functor. Stereotype spaces satisfy a reasonable completeness-type condition (each precompact set is relatively compact). The paper provides a systematic study of stereotype locally convex spaces, algebras, and modules. Sections 1-9 are devoted to stereotype spaces. The author introduces the operations of saturation, pseudosaturation, and pseudocompletion of a locally convex space, and studies their basic properties. He gives examples of stereotype spaces and studies relations between various properties of a stereotype space and its dual. Next he considers spaces of linear and bilinear mappings and introduces stereotype versions of projective and injective tensor products. The author discusses integration theory for functions with values in stereotype spaces, and applies this theory to tensor products of stereotype spaces consisting of functions, measures, and distributions. Section 9 is devoted to the stereotype approximation property. Homological characterizations of the stereotype approximation property are given. The results presented in these fir…
reviewed by: José Bonet
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 46H05
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46-02
zbMATH DE Number: 1981995
zbMATH Keywords: locally convex spaces
zbMATH Keywords: topological algebras
zbMATH Keywords: monoidal categories
Wikidata QID: Q55934106
author: Sergei S. Akbarov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1981995