Pontryagin duality for spaces of continuous functions
type: Item
instance of: scholarly article
title: Pontryagin duality for spaces of continuous functions
zbMATH Open document ID: 0942.54013
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-08-28T00:00:00Z
review text: Let \(X\) be a completely regular Hausdorff space and \(C_p(X)\) denote the space of continuous real valued functions on \(X\) with the topology of pointwise convergence. We know that if \(X\) is not discrete, then \(C_p(X)\) is not reflexive as a topological linear space. The authors regard \(C_p(X)\) instead as a topological abelian group \(G\), take the dual group of characters with the compact open topology, and say that \(G\) is \(P\)-reflexive in case the natural homomorphism from \(G\) to its second dual space is a topological isomorphism. They give an example where \(X\) is not discrete and \(C_p(X)\) is \(P\)-reflexive, and they show that if \(X\) is metrizable, then \(C_c(X)\) (compact open topology instead of pointwise convergence) is \(P\)-reflexive. They show, among other things, that if \(X\) is an uncountable discrete space together with a point at infinity whose neighborhoods are complements of countable subsets, then the question of whether or not \(C_p(X)\) is \(P\)-reflexive is undecidable in ZFC.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 46E10
zbMATH DE Number: 1423626
zbMATH Keywords: \(P\)-space
zbMATH Keywords: caliber
zbMATH Keywords: \(P\)-reflexive
author: Salvador Hernández
author: Vladimir V. Uspenskij
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6627
OpenAlex ID: W2085278602
cites work: Q4220648
cites work: Additive subgroups of topological vector spaces
cites work: Q5683258
cites work: Pontryagin-van Kampen reflexivity for free Abelian topological groups
cites work: Q5655867
cites work: Q5723435
cites work: Extensions of the Pontrjagin duality. I: Infinite products
cites work: Extensions of the Pontrjagin duality. II: Direct and inverse sequences
cites work: Q3724944
cites work: TOPOLOGICAL VECTOR SPACES OF CONTINUOUS FUNCTIONS
cites work: Q3709279
cites work: Q5830010
cites work: Sequential Compactness of X Implies a Completeness Property for C(X)
cites work: Q5613420
cites work: On locally convex vector spaces of continuous functions
cites work: The Pontrjagin duality theorem in linear spaces
DOI: 10.1006/JMAA.1999.6627
description: scientific article; zbMATH DE number 1423626