## 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
Full Text:

### References:

  Bennett, G., Boos, J. and T. Leiger: Sequences of 0’s and 1’s. Studia Math. 149 (2002), 75 - 99. · Zbl 0995.46010  Bennett, G. and N. J. Kalton: Inclusion theorems for K-spaces. Canad. J. Math. 25 (1973), 511 - 524. · Zbl 0272.46009  Boos, J.: Classical and Modern Methods in Summability. Oxford: Oxford Univ. Press 2000. · Zbl 0954.40001  Boos, J. and M. R. Parameswaran: The potency of weighted means: Addendum to a paper of Kuttner and Rarameswaran. J. Anal. 7 (1999), 219 - 224. · Zbl 0957.40003  Drewnowski, L. and P. J. Paúl: The Nikodym property for ideals of sets defined by matrix summability methods. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000), 485-503, Perspectives in mathematical analysis (Spanish). · Zbl 1278.46002  Kuttner, B. and M. R. Parameswaran: A class of weighted means as potent concervative methods. J. Anal. 4 (1996), 161 - 172. · Zbl 0880.40003  Rath, D.: On conservative nonregular Nörlund and weighted mean matrices. J. Anal. 6 (1998), 13 - 18. · Zbl 0922.40009  Stuart, C. E.: A generalization of the Nikodym Boundedness Theorem.Preprint.  Wilansky, A.: Modern Methods in Topological Vector Spaces. New York: McGraw-Hill 1978. · Zbl 0395.46001  Wilansky, A.: Summability through Functional Analysis (Notas de Matemática: Vol. 85). Amsterdam et al.: North Holland 1984. · Zbl 0531.40008  Zeltser, M., Boos, J. and T. Leiger: Sequences of 0’s and 1’s: New results via double sequence spaces. J. Math. Anal. Appl. 275 (2002), 883 - 899. · Zbl 1037.46007
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.