Sequences of 0's and 1's: new results via double sequence spaces.
type: Item
instance of: scholarly article
title: Sequences of 0's and 1's: new results via double sequence spaces.
zbMATH Open document ID: 1037.46007
DOI: 10.1016/S0022-247X(02)00444-4
author: Maria Zeltser
author: Johann Boos
author: Toivo Leiger
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: This paper continues the joint investigation by \textit{G. Bennett, J. Boos} and \textit{T. Leiger} [Stud. Math. 149, 75--99 (2002; Zbl 0995.46010)] of the extent to which sequence spaces are determined by the sequences of 0's and 1's that they contain. The first main result gives a negative answer to question 6 in their paper. There exists a sequence space \(E\) such that each matrix domain containing all sequences of zeros and ones in \(E\) contains all of \(E\), but such that this statement fails if we replace matrix domains by separable FK-spaces. The second main result starts from Hahn's theorem that tells us that each matrix domain including \(X\), the set of all sequences of 0's and 1's, contains all bounded sequences. It is shown that there exists a really `small' subset \(\widetilde X\) of \(X\) such that Hahn's theorem remains true when \(X\) is replaced with it. The proofs of both results have in common that, by identifying sequence spaces and double sequence spaces, the constructions and the required investigations are done in double sequence spaces that allow the description of finer structures.
reviewed by: Babban Prasad Mishra
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 40A05
zbMATH DE Number: 1866723
zbMATH Keywords: Schur's theorem
zbMATH Keywords: inclusion theorems
zbMATH Keywords: sequences of zeros and ones
zbMATH Keywords: dense subspaces of \(\ell^\infty\)
zbMATH Keywords: matrix Hahn property
zbMATH Keywords: separable Hahn property
zbMATH Keywords: double sequence spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1866723