\(k\)-spaces and duals of non-Archimedean metrizable locally convex spaces
type: Item
instance of: scholarly article
title: \(k\)-spaces and duals of non-Archimedean metrizable locally convex spaces
zbMATH Open document ID: 1405.46056
author: Maria Cristina Perez-Garcia
published in: Forum Mathematicum
publication date: 2018-10-04T00:00:00Z
review text: A Hausdorff topological space \(X\) is called a \(k\)-space if a subset \(A\) of \(X\) is open (closed) if and only if \( A\cap C\) is open (closed) in \(C\) for every compact subset \(C\) of \(X\). If \(X\) is zero-dimensional, i.e., the topology admits a base formed of clopen (closed and open) sets, then it is called a \(k_0\)-space if and only if the above property holds for any clopen subset \(A\) of \(X\). The paper is concerned with the \(k\)-space properties of non-Archimedean locally convex spaces. Let \(\mathbb{K}\) be a complete non-Archimedean valued field and \(E\) a non-Archimedean locally convex space over \(\mathbb{K}\), with dual \(E'\) (see [\textit{C. Perez-Garcia} and \textit{W. H. Schikhof}, Locally convex spaces over non-Archimedean valued fields. Cambridge: Cambridge University Press (2010; Zbl 1193.46001)]). A subset \(A\) of \(E\) is called a compactoid if, for every neighborhood \(V\) of \(0\in E\), there exists a finite set \(Z\subset E\) such that \(A\subset V+\)aco\((Z)\), where aco\((Z)\) denotes the absolutely convex hull of \(Z\). One denotes by \(\sigma(E',E)\) (\(c(E',E)\)) the locally convex topology on \(E'\) of uniform convergence on all finite (compactoid) subsets of \(E\) and let \(E'_c\) be the space \(E'\) equipped with the topology \(c(E',E)\). If \(E\) is a metrizable and \(\mathbb{K}\) is locally compact, then \(E'_c\) is a \(k\)-space, hence a \(k_0\)-space. This result is no longer true if \(\mathbb{K}\) is not locally compact -- in this case, the space \(\left(\mathbb{K}^{\mathbb{N}}\right)'_c\) is not a \(k\)-space. In fact, the local compactness of \(\mathbb{K}\) is equivalent to the fact that, for every metrizable locally convex space \(E\) over \(\mathbb{K}\), the space \(E'_c\) is a \(k\)-space. The author also proves a non-Archimedean analog of a result of Dieudonné -- if \(E\) is metrizable and of countable type, then \(E'_c\) is a \(k\)-space if and only if \(c(E',E)\) is the finest topology on \(E'\) agreeing wi…
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 54D50
zbMATH DE Number: 6947026
zbMATH Keywords: \( k\)-space
zbMATH Keywords: \(k_0\)-space
zbMATH Keywords: non-Archimedean valued field
zbMATH Keywords: non-Archimedean locally convex space
zbMATH Keywords: non-Archimedean Banach-Dieudonné theorem
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: forum-2016-0080
OpenAlex ID: W2883620157
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DOI: 10.1515/FORUM-2016-0080
description: scientific article; zbMATH DE number 6947026