## 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.

### MSC:

 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:

### References:

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