## Sequences of 0’s and 1’s: sequence spaces with the separable Hahn property.(English)Zbl 1151.46003

Given a sequence space $$E$$, denote by $$\chi(E)$$ the linear span of the sequences of 0’s and 1’s contained in $$E$$. We say that $$E$$ has the Hahn property, the separable Hahn property, or the matrix Hahn property if the implication $$\chi(E)\subseteq F\Rightarrow E\subseteq F$$ holds whenever $$F$$ is an FK-space, a separable FK-space, or a matrix domain $$c_A$$, respectively.
Theorem 5.2 of [G. Bennett, J. Boos and T. Leiger, Studia Math. 149, No. 1, 75–99 (2002); addendum ibid. 171, No. 3, 305–309 (2005; Zbl 0995.46010)] states that for a monotone sequence space $$E$$ containing $$\varphi$$, (i) the matrix Hahn property, (ii) the separable Hahn property and (iii) the equality $$\chi(E)^{\beta}=E^{\beta}$$ are equivalent. However, as it was pointed out in the addendum, there is a gap in the proof of the implication (iii)$${}\Rightarrow{}$$(ii) of this result.
This ‘failed’ result was applied in the paper to three different instances, namely, the space $$|ac|_0$$ of strongly almost-null sequences, the space $$\ell^\infty(|\lambda|)$$ where $$\lambda$$ is a lacunary sequence of indices, and the space $$\ell^\infty \cap z^\alpha$$ where $$z \in c_0 \backslash \ell^1$$ (see any of the papers for the precise definitions).
C. E. Stuart and P. Abraham [J. Math. Anal. Appl. 300, No. 2, 351–361 (2004; Zbl 1081.28005)] proved that for the space $$|ac|_0$$ the implication (iii)$${}\Rightarrow{}$$(ii) holds true, and the purpose of the paper [J. Boos and T. Leiger, Acta Math. Hung. 115, No. 4, 341–356 (2007; Zbl 1151.46002), reviewed above] is to show, through specific gliding hump proofs that do not apply in general, that this is also the case for the remaining instances $$\ell^\infty(|\lambda|)$$ and $$\ell^\infty \cap z^\alpha$$.
The main result of the paper under review is a counterexample showing that the implication (iii)$${}\Rightarrow{}$$(ii) does not hold in general for monotone sequence spaces. Also, a number of new examples of spaces with the separable Hahn property are provided.

### MSC:

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

### Citations:

Zbl 0995.46010; Zbl 1081.28005; Zbl 1151.46002
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