## Sequences of 0’s and 1’s. Classes of concrete ‘big’ Hahn spaces.(English)Zbl 1061.46004

Let $$\omega$$ denote the space of all real sequences $$x= (x_k)$$ $$(k\in\mathbb{N})$$. The space $$\omega$$, supplied with the topology of coordinatewise convergence, is an FK-space, i.e., a sequence space endowed with a complete, metrizable l.c. topology such that all the mappings $$x\to x_k$$ $$(k\in\mathbb{N})$$ are continuous. A normable FK-space is called a BK-space.
Any vector subspace of $$\omega$$ is called a sequence space. A sequence space can have at most one FK-topology.
Let $$\chi$$ be the set of all sequences of $$0$$’s and $$1$$’s and, for any sequence space $$E$$, let $$\chi(E)$$ denote the linear hull of the sequences of $$0$$’s and $$1$$’s contained in $$E$$. It is always true that $$\chi(E)\subset\chi(\ell^\infty)\cap E$$. For any infinite matrix $$B= (b_{nk})$$ with real entries, the summability domain $$c_B$$ of $$B$$ is defined as $c_B= \biggl\{x= (x_k)\in \omega\mid Bx:= \Bigl(\sum_k b_{nk} x_k\Bigr)_n\in c\biggr\}$ ($$Bx$$ is assumed to be well-defined). Let $$\lim_B$$ denote the mapping $$\lim_B: c_B\to\mathbb{R}$$, $$x\mapsto\lim Bx$$. Then the pair $$(c_B,\lim_B)$$ is called a matrix method. $$c_B$$ is a separable FK-space and, if the matrix method is conservative, i.e., $$c\subset c_B$$, its bounded domain $$\ell^\infty\cap c_B$$ is a closed subspace of $$\ell^\infty$$.
A sequence space $$E$$ is said to have the Hahn property if for any FK-space $$F$$ we have $\chi(E)\subset F\Rightarrow E\subset F.$ In this case, $$E$$ is called a Hahn space. If, moreover, $$F$$ is a separable FK-space, then $$E$$ is said to have the separable Hahn property. Finally, $$E$$ has the matrix Hahn property if for a matrix method $$(c_B,\lim_B)$$ we have $$\chi(E)\subset c_B\Rightarrow E\subset c_B$$.
It has been proved [G. Bennett and N. J. Kalton, Can. J. Math. 25, 511–524 (1973; Zbl 0272.46009)] that $$E$$ has the Hahn property if and only if $$\chi(E)$$ is a dense barrelled subspace of $$E$$. Also, G. Bennett, J. Boos and T. Leiger [Stud. Math. 149, No. 1, 75–99 (2002; Zbl 0995.46010)] have shown that for a sequence space $$E$$, the inclusion $$bs+ \text{Sp}\{e\}\subset E\subset\ell^\infty$$ implies that $$E$$ has the Hahn property. (As usual, $$bs= \{x= (x_k)\in\omega\mid\sup_n|\sum^n_{k=1} x_k|< \infty\}$$.)
In this paper, the authors continue the joint investigation of Bennett, Boos and Leiger (loc. cit.) and M. Zeltser, J. Boos and T. Leiger [J. Math. Anal. Appl. 275, No. 2, 883-899 (2002; Zbl 1037.46007)] of the extent to which sequence spaces are determined by the sequences of $$0$$’s and $$1$$’s that they contain. For any partition $$N=(N_i)$$ of $$\mathbb{N}$$ into disjoint infinite subsets of $$\mathbb{N}$$, they define $bs(N)= \Bigl\{x\in\omega\mid\| x\|_{bs(N)}:= \sup_j\|(x_k)_{k\in N_j}\|_{bs}< \infty\Bigr\}$ and show that the result of Bennett et al. (loc cit.) remains true if the space $$bs$$ is replaced by $$bs(N)$$. (Note that if $$N$$ is finite then $$bs(N)$$ is a subspace of $$bs$$.)
As an application of the above result, they prove that the bounded domain of a weighted mean method (with positive weights) is a Hahn space if and only if the diagonal of the matrix of the method is a null sequence, and a similar result applies to the bounded domains of regular Nörlund methods. Combining their results with the Bennett-Kalton result, they deduce that a large class of concrete closed subspaces $$E$$ of $$\ell^\infty$$ such that $$\chi(E)$$ is a dense barrelled subspace can be identified by simple conditions. The paper ends with two open problems.

### MSC:

 46A45 Sequence spaces (including Köthe sequence spaces) 46A35 Summability and bases in topological vector spaces 40C05 Matrix methods for summability 40H05 Functional analytic methods in summability

### Citations:

Zbl 0272.46009; Zbl 0995.46010; Zbl 1037.46007
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### References:

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