##
**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.

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.

Reviewer: Constantine G. Lascarides (Athens)

### 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 |

### Keywords:

dense barrelled subspaces; Hahn space (property); separable Hahn property; matrix Hahn property; bounded domains; weighted mean method; Nörlund method
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\textit{J. Boos} and \textit{M. Zeltser}, Z. Anal. Anwend. 22, No. 4, 819--842 (2003; Zbl 1061.46004)

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### References:

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[2] | Bennett, G. and N. J. Kalton: Inclusion theorems for K-spaces. Canad. J. Math. 25 (1973), 511 - 524. · Zbl 0272.46009 |

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[8] | Stuart, C. E.: A generalization of the Nikodym Boundedness Theorem.Preprint. |

[9] | Wilansky, A.: Modern Methods in Topological Vector Spaces. New York: McGraw-Hill 1978. · Zbl 0395.46001 |

[10] | Wilansky, A.: Summability through Functional Analysis (Notas de Matemática: Vol. 85). Amsterdam et al.: North Holland 1984. · Zbl 0531.40008 |

[11] | 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 |

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