Finest chains of subspaces in infinite-dimensional vector spaces
type: Item
instance of: scholarly article
title: Finest chains of subspaces in infinite-dimensional vector spaces
zbMATH Open document ID: 0882.15001
author: Martin Menth
published in: Mathematische Semesterberichte
publication date: 1997-07-17T00:00:00Z
review text: Suppose that \(K\) is a set of subspaces of a vector space \(V\), such that \(\{0\}\), \(V\in K\) and such that \(K\) is totally ordered by inclusion. Such a \(K\) is called a chain of subspaces. A finest chain \(K\) is maximal, i.e. no subspaces can be added to \(K\) without destroying the total order. Clearly, if \(V\) is \(d\)-dimensional \((d<\aleph_0)\), then each finest chain has cardinality \(d+1\). It is shown that this result remains true for \(d=\aleph_0\), but fails to hold in case of uncountable dimension. For example, if \(F\) is a field, then the vector space of all mappings \(\mathbb{N}\to F\) has no countable basis, but a countable finest chain of subspaces.
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 1036098
zbMATH Keywords: vector space
zbMATH Keywords: chain of subspaces
zbMATH Keywords: dimension
reviewed by: Hans Havlicek
MaRDI profile type: Publication
full work available at URL: s005910050017
OpenAlex ID: W2030029016
DOI: 10.1007/S005910050017
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description: scientific article; zbMATH DE number 1036098