Pontryagin duality for subgroups and quotients of nuclear spaces
type: Item
instance of: scholarly article
title: Pontryagin duality for subgroups and quotients of nuclear spaces
zbMATH Open document ID: 0593.46006
DOI: 10.1007/BF01472136
author: Wojciech Banaszczyk
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164078
review text: Let G be an abelian topological group and \(G^\wedge\) the dual group, i.e. the group of all continuous characters (homomorphisms into R/Z) of G, endowed with the topology of uniform convergence on precompact sets. A subgroup H of G is said to be dually closed if \((G/H)^\wedge\) separates the points of G/H and dually embedded if each \(\chi\in H^\wedge\) can be extended to some \({\tilde \chi}\in G^\wedge\). We say that G is reflexive if the canonical embedding \(G\to G^{\wedge\wedge}\) is a topological isomorphism. A reflexive group G is called strongly reflexive if the duality between G and \(G^\wedge\) induces dualities between appropriate subgroups and quotients of G and \(G^\wedge\). In other words, G is strongly reflexive if every closed subgroup of G and of \(G^\wedge\) is both dually closed and dually embedded and if all closed subgroups and Hausdorff quotient groups of G and of \(G^\wedge\) are reflexive. The following two results are obtained: 1) If K is an additive subgroup of a nuclear space E, then every equicontinuous subset of \(K^\wedge\) is the canonical image of an equicontinuous subset of \(E^\wedge\). In particular, K is dually embedded. 2) Every nuclear Fréchet space is strongly reflexive as an additive topological group. This can be extended to some wider class of groups, e.g. to countable products of nuclear Fréchet spaces and LCA groups.
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
Mathematics Subject Classification ID: 46A20
Mathematics Subject Classification ID: 22D35
zbMATH DE Number: 3953539
zbMATH Keywords: Pontryagin duality
zbMATH Keywords: dual group
zbMATH Keywords: topology of uniform convergence on precompact sets
zbMATH Keywords: dually closed
zbMATH Keywords: dually embedded
zbMATH Keywords: equicontinuous subset
zbMATH Keywords: Every nuclear Fréchet space is strongly reflexive as an
zbMATH Keywords: additive topological group
zbMATH Keywords: countable products of nuclear Fréchet spaces
zbMATH Keywords: Every nuclear Fréchet space is strongly reflexive as an additive topological group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3953539