On property \(K\) in \(F\)-spaces (Q2702786)

The authors say that a topological linear space \(X\) is a \(K\)-space or has property \(K\) if every sequence \(\{ \xi_n\} \) in \(X\) with \(\xi_n \to 0\) contains a subsequence \(\{ \xi_{p_n}\} \) such that the series \(\sum_{n=1}^{\infty } \xi_{p_n}\) is convergent. (This completeness type property was first considered by S. Mazur and W. Orlicz in 1953.) The main result of the paper is NEWLINENEWLINENEWLINETheorem 1: Let \(X\) be an \(F\)-space with dim \(X=\mathfrak c\), let \(E\) be an \(F_{\sigma}\)-subspace of \(X\) with \(\dim E=\mathfrak c\) and codim \(E\geqslant \aleph_0\) and let \(F\) be a \(K_\sigma \)-subspace of \(X\) with \(E\cap F=\{0\}\). Then for every subspace \(H\) of \(X\) with \(\text{dim }H<\mathfrak c\) and \((E\oplus F)\cap H = \{0\}\) there exist dense \(K\)-subspaces \(Y_1\) and \(Y_2\) of \(X\) such that \(E\oplus Y_1 = E\oplus Y_2 = X\) and \(Y_1 \cap Y_2 = F\oplus H\). NEWLINENEWLINENEWLINEThe subspaces \(Y_1\) and \(Y_2\) are constructed by transfinite induction, using techniques developed by \textit{J. Burzyk} in a earlier paper [``Decompositions of \(F\)-spaces into spaces with properties \(K\), \(N\), or \(\kappa \)'', in: Generalized functions and convergence, Katowice, 1988, 317-329, World Sci. Publishing, Teaneck, NJ (1990)] for a related purpose. NEWLINENEWLINENEWLINEReviewer's remarks: NEWLINENEWLINENEWLINE(1) In the formulation of Theorem 3 the assumption that dim \(H=\mathfrak c\) has been inadvertently omitted. NEWLINENEWLINENEWLINE(2) The proof of Theorem 1 needs some corrections. Namely, in \((e_1)\) the second inequality should be ``\(\leq \)'', and in (20) ``\(E\oplus \widetilde F\oplus \widetilde H_{\alpha_0}\)'' should be ``\(E\oplus \widetilde F +\widetilde H_{\alpha_0}\)''. Moreover, the definitions of \(\xi^i\) and \(\omega^i\) should be modified to guarantee \((e_3)\). Finally, in the second implication at the bottom of p. 219, the last ``\(<\)'' should apparently be ``\(\neq \)''. NEWLINENEWLINENEWLINE(3) For remarks on Burzyk's paper referred to above and, in particular, its relation to a paper by \textit{L. Drewnowski} and the reviewer [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 28, No. 2, 175-188 (1989; Zbl 0770.46001], see MR 92b:46007. In this connection, Theorem 1 of that paper is better suited for the proof of Lemma 6 of the paper under review than Lemma 1.