JournalszaaVol. 22, No. 4pp. 819–842

Sequences of 0's and 1's. Classes of Concrete `big' Hahn Spaces

  • Johann Boos

    Fernuniversität-GHS, Hagen, Germany
  • Maria Zeltser

    Tartu University, Estonia
Sequences of 0's and 1's. Classes of Concrete `big' Hahn Spaces cover
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Abstract

This paper continues the joint investigation by G. Bennett, J. Boos and T. Leiger [Studia Math. 149 (2002) 75--99] and J. Boos, T. Leiger and M. Zeltser [J. Math. Anal. Appl. 275 (2002) 883--899] of the extent to which sequence spaces are determined by the sequences of 00's and 11's that they contain. Bennett et al. proved that each subspace EE of \ell^\infty containing the sequence e=(1,1,)e = (1,1, \dots) and the linear space bsbs of all sequences with bounded partial sums is a Hahn space, that is, an FK-space FF contains EE whenever it contains (the linear hull χ(E)\chi(E) of) the sequences of 00's and 11's in EE. In some sense these are `big' subspaces of \ell^\infty. Theorem 2.6, one of the main results of this paper, tells us that this result remains true if we replace bsbs with suitably defined spaces bs(N)bs(N) which are subspaces of bsbs when NN is a finite partition of N\Bbb N. As an application of the main result, two large families of closed subspaces EE of \ell^\infty being Hahn spaces are presented: The bounded domain EE of a weighted mean method (with positive weights) is a Hahn space if and only if the diagonal of the matrix defining the method is a null sequence; a similar result applies to the bounded domains of regular N\"{o}rlund methods. Since an FK-space EE is a Hahn space if and only if χ(E)\chi(E) is a dense barrelled subspace of EE, by these results, a large class of concrete closed subspaces EE of \ell^\infty such that χ(E)\chi(E) is a dense barrelled subspace can be identified by really simple conditions. A further application gives a negative answer to Problem 7.1 in the paper mentioned above.

Cite this article

Johann Boos, Maria Zeltser, Sequences of 0's and 1's. Classes of Concrete `big' Hahn Spaces. Z. Anal. Anwend. 22 (2003), no. 4, pp. 819–842

DOI 10.4171/ZAA/1175