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Bases and closures under infinite sums
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Bruhn, Henning and Georgakopoulos, Agelos (2011) Bases and closures under infinite sums. Linear Algebra and Its Applications, Vol.435 (No.8). pp. 2007-2018. doi:10.1016/j.laa.2011.03.029 ISSN 0024-3795.
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Official URL: http://dx.doi.org/10.1016/j.laa.2011.03.029
Abstract
Motivated by work of Diestel and Kühn on the cycle spaces of infinite graphs we study the ramifications of allowing infinite sums in a module RM. We show that every generating set in this setup contains a basis if the ground set M is countable, but not necessarily otherwise. Given a family N⊆RM, we determine when the infinite-sum span N of N is closed under infinite sums, i.e. when N=N. We prove that this is the case if R is a field or a finite ring and each element of M lies in the support of only finitely many elements of N. This is, in a sense, best possible. We finally relate closures under infinite sums to topological closures in the product space RM.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Linear Algebra and Its Applications | ||||
Publisher: | Elsevier Inc | ||||
ISSN: | 0024-3795 | ||||
Official Date: | 2011 | ||||
Dates: |
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Volume: | Vol.435 | ||||
Number: | No.8 | ||||
Page Range: | pp. 2007-2018 | ||||
DOI: | 10.1016/j.laa.2011.03.029 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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