Galois connections between generating systems of sets and sequences
type: Item
instance of: scholarly article
title: Galois connections between generating systems of sets and sequences
zbMATH Open document ID: 1387.46022
author: Rauni Lillemets
published in: Positivity
publication date: 2017-12-06T00:00:00Z
review text: For a Banach space \(X\), consider the collection \(\mathbf{K}\) of all compact sets of \(X\) and the collection \(\mathbf{c}\) of all convergent sequences of \(X\). Obviously, we may characterize \(\mathbf{K}\) in terms of \(\mathbf{c}\): A set \(K\) is in \(\mathbf{K}\) if every sequence of \(K\) has a subsequence which belongs to \(\mathbf{c}\). This way to generate a collection of sets from a collection of sequences was first studied by [\textit{I. Stephani}, Math. Nachr. 99, 13--27 (1980; Zbl 0474.47019)], where she introduced the notions of generating systems of sets (\(\mathrm{GSet}\)) and generating systems of sequences (\(\mathrm{GSeq}\)). This construction gives an operator \(\Psi: \mathrm{GSeq}\rightarrow \mathrm{GSet}\). In the article under review, the author takes up Stephani's work and defines the operator \(\Phi: \mathrm{GSet}\rightarrow \mathrm{GSeq}\) as follows: for \(\mathbf{G} \in \mathrm{GSet}\), a sequence belongs to \(\Phi(\mathbf{G})\) if and only if it is contained in some \(G \in \mathbf{G}\). Since \(\mathrm{GSet}\) is ordered by the inclusion, Stephani defined in \(\mathrm{GSeq}\) a preorder. The author shows that this preorder leads to an equivalence relation \(\sim\) on \(\mathrm{GSeq}\) and induces an order on \(\mathrm{GSeq}/{\sim}\). The saturated systems of sequences, \(\mathrm{satGSeq}\), of all collection of sequences \(\mathbf g\) is considered such that \(\mathbf{g}=\Phi \circ \Psi(\mathbf{g})\) and the sequentially generatable systems of sets, \(\mathrm{seqGSet}\), as all \(\mathbf{G} \in \mathrm{GSet}\) such that \(\mathbf{G}=\Phi(\mathbf{g})\) for some \(\mathbf{g} \in \mathrm{GSeq}\). Then the operators \(\phi: \mathrm{GSet}\rightarrow \mathrm{GSeq}/{\sim}\) and \(\psi: \mathrm{GSeq}/{\sim}\rightarrow \mathrm{GSet}\) are defined as \[ \phi(\mathbf{G})=[\Phi(\mathbf{G}] \quad \text{and} \quad \psi([\mathbf{g}])=\Psi(\mathbf{g}), \] and it is shown that the pair \((\phi,\psi)\) is a Galois connection between the ordered sets …
reviewed by: Pablo Turco
Mathematics Subject Classification ID: 46B99
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 47L20
zbMATH DE Number: 6816285
zbMATH Keywords: Banach spaces
zbMATH Keywords: operator ideals
zbMATH Keywords: generating systems of sets
zbMATH Keywords: generating systems of sequences
zbMATH Keywords: Galois connection
zbMATH Keywords: lattices
MaRDI profile type: Publication
full work available at URL: s11117-016-0451-z
OpenAlex ID: W2534031517
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DOI: 10.1007/S11117-016-0451-Z
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description: scientific article; zbMATH DE number 6816285