Duality of topological modules over normed rings
type: Item
instance of: scholarly article
title: Duality of topological modules over normed rings
zbMATH Open document ID: 1398.46039
author: Hailan Liang
author: Shu-Qi Huang
published in: Topology and its Applications
publication date: 2018-06-07T00:00:00Z
review text: The paper studies topological (left or right) modules over unital normed rings. Let \(M, N\) be two topological left modules over a unital normed ring \(R\). Then the set (actually, a topological group under the pointwise addition) of all continuous \(R\)-homomorphisms from \(M\) to \(N\) is denoted by \({\mathcal B}(M, N)\). In the paper, it is shown that, if \(M\) and \(N\) are locally bounded and \(N\) is Hausdorff, then \({\mathcal B}(X, Y)\) is a locally bounded Hausdorff space. Let \(R\) be a unital topological ring and \(S\) an admissible ring. In this case, the functors \({\mathcal B}(N, -)\) and \({\mathcal B}(-, N)\) are studied for a topological \((R, S)\)-bimodule \(N\). It is shown that, for any topological left \(R\)-module \(M\), the set \({\mathcal B}(N, M)\) is a topological left \(S\)-module and the set \({\mathcal B}(M, N)\) is a topological right \(S\)-module. Some properties of topologically (left or right) exact sequences are also proved for admissible rings.
reviewed by: Mart Abel
Mathematics Subject Classification ID: 46H25
Mathematics Subject Classification ID: 46M15
Mathematics Subject Classification ID: 46M18
zbMATH DE Number: 6882112
zbMATH Keywords: topological module
zbMATH Keywords: normed ring
zbMATH Keywords: duality
zbMATH Keywords: topologically (left or right) exact sequences
zbMATH Keywords: operator topology
MaRDI profile type: Publication
full work available at URL: j.topol.2018.05.006
OpenAlex ID: W2803235669
cites work: Q4023148
cites work: Q3759347
cites work: Note on the open mapping theorem
cites work: Approximate identities and factorization in Banach modules
cites work: Pontryagin Duality for Topological Modules
cites work: Q3401901
cites work: Group duality with the topology of precompact convergence
cites work: Q4743319
cites work: Q4340096
cites work: Q5620935
cites work: Q3992929
cites work: Q3135126
DOI: 10.1016/J.TOPOL.2018.05.006
description: scientific article; zbMATH DE number 6882112