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Sequences of 0’s and 1’s: new results via double sequence spaces. (English) Zbl 1037.46007

This paper continues the joint investigation by G. Bennett, J. Boos and 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.

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40A05 Convergence and divergence of series and sequences

Citations:

Zbl 0995.46010
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References:

[1] Bennett, G.; Boos, J.; Leiger, T., Sequences of 0’s and 1’s, Studia math., 149, 75-99, (2002) · Zbl 0995.46010
[2] Bennett, G.; Kalton, N.J., Inclusion theorems for K-spaces, Canad. J. math., 25, 511-524, (1973) · Zbl 0272.46009
[3] Boos, J., Classical and modern methods in summability, (2000), Oxford Univ. Press Oxford · Zbl 0954.40001
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[5] Hahn, H., Über folgen linearer operationen, Monatsh. math., 32, 3-88, (1922) · JFM 48.0473.01
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