On property \(K\) in \(F\)-spaces. (English) Zbl 0961.46002

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
Theorem 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 \(Y_1 \cap Y_2 = F\oplus H\).
The subspaces \(Y_1\) and \(Y_2\) are constructed by transfinite induction, using techniques developed by 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.
Reviewer’s remarks:
(1) In the formulation of Theorem 3 the assumption that dim \(H=\mathfrak c\) has been inadvertently omitted.
(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 “\(\neq \)”.
(3) For remarks on Burzyk’s paper referred to above and, in particular, its relation to a paper by 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.


46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
40A05 Convergence and divergence of series and sequences


Zbl 0770.46001
Full Text: EuDML


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