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

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 $0$'s and $1$'s that they contain. Bennett et al. proved that each subspace $E$ of ${\ell }^{\infty }$ containing the sequence $e=(1,1,\dots )$ and the linear space $bs$ of all sequences with bounded partial sums is a Hahn space, that is, an FK-space $F$ contains $E$ whenever it contains (the linear hull $\chi (E)$ of) the sequences of $0$'s and $1$'s in $E$. 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 $bs$ with suitably defined spaces $bs(N)$ which are subspaces of $bs$ when $N$ is a finite partition of $\mathbb{N}$. As an application of the main result, two large families of closed subspaces $E$ of ${\ell }^{\infty }$ being Hahn spaces are presented: The bounded domain $E$ 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 Nörlund methods. Since an FK-space $E$ is a Hahn space if and only if $\chi (E)$ is a dense barrelled subspace of $E$, by these results, a large class of concrete closed subspaces $E$ of ${\ell }^{\infty }$ such that $\chi (E)$ 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.