Sequences of 0's and 1's: new results via double sequence spaces

Abstract

This paper continues the joint investigation by Bennett et al. (2001) 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 of the 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 goes on from Hahn's theorem that tells us that each matrix domain including χ, the set of all sequences of 0's and 1's, contains all of the bounded sequences: It is shown that there exists a really ‘small’ subset χ of χ such that Hahn's theorem remains true when χ 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.