Addendum to: ``Sequences of 0's and 1's'' (Studia Math. 149 (2002), 75–99)

Abstract

Abstract. There is a nontrivial gap in the proof of Theorem 5.2 of [2] which is one of the main results of that paper and has been applied three times (cf. [2, Theorem 5.3, (G) in Section 6, Theorem 6.4]). Till now neither the gap has been closed nor a counterexample found. The aim of this paper is to give, by means of some general results, a better understanding of the gap. The proofs that the applications hold will be given elsewhere.Concerning notations and preliminary results we refer to the original paper [2] and to [9], [10] and [3]. Let χ be the set of all sequences of 0's and 1's, and, if E is any sequence space, let χ(E) denote the linear hull of χ ∩ E.In [2] (cf. also [11,4]) the authors considered sequence spaces E with the property that (i) E has the matrix Hahn property; (ii) E has the separable Hahn property; (iii) χ(E) β = E β .

Sequences of 0's and 1's: sequence spaces with the separable Hahn property

“…In [3] it was pointed out that the paper [1] by G. Bennett, J. Boos and T. Leiger contains a nontrivial gap in the proof of Theorem 5.2. This theorem is one of the main results of the paper and it was applied three times (cf.…”

“…In [3] it was discovered that one of the main results in [1] (Theorem 5.2), applied to three spaces, contains a nontrivial gap in the argument, but neither the gap was closed nor a counterexample was provided. In [4] the authors verified that all three above mentioned applications of the theorem are true and stated a problem concerning the topological structure of one of these three spaces.…”