##
**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.

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.

Reviewer: Babban Prasad Mishra (Gorakhpur)

### MSC:

46A45 | Sequence spaces (including Köthe sequence spaces) |

40A05 | Convergence and divergence of series and sequences |

### Keywords:

Schur’s theorem; inclusion theorems; sequences of zeros and ones; dense subspaces of \(\ell^\infty\); matrix Hahn property; separable Hahn property; double sequence spaces### Citations:

Zbl 0995.46010
PDF
BibTeX
XML
Cite

\textit{M. Zeltser} et al., J. Math. Anal. Appl. 275, No. 2, 883--899 (2002; Zbl 1037.46007)

Full Text:
DOI

### 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 |

[4] | Boos, J.; Leiger, T.; Zeller, K., Consistency theory for SM-methods, Acta math. hungar., 76, 83-116, (1997) · Zbl 0930.40001 |

[5] | Hahn, H., Über folgen linearer operationen, Monatsh. math., 32, 3-88, (1922) · JFM 48.0473.01 |

[6] | Wilansky, A., Modern methods in topological vector spaces, (1978), McGraw-Hill New York · Zbl 0395.46001 |

[7] | Wilansky, A., Summability through functional analysis, North-holland math. stud., 85, (1984), North-Holland Amsterdam · Zbl 0531.40008 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.