×

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

Citations:

Zbl 0770.46001
PDF BibTeX XML Cite
Full Text: EuDML

References:

[1] ALEXIEWICZ A.: On sequences of operations, (II). Studia Math. 11 (1950), 200-236. · Zbl 0039.12002
[2] ANTOSIK P.-BURZYK J.: Sequential conditions for barrelledness and bornology. Bull. Polish Acad. Sci. Math. 35 (1987), 151-155. · Zbl 0642.46003
[3] ANTOSIK P.-SWARTZ C.: Matrix Methods in Analysis. Lecture Notes in Math. 1113, Springer-Verlag, Berlin, 1985. · Zbl 0564.46001
[4] BURZYK J.: An example of a non-complete normed N-space. Bull. Polish Acad. Sci. Math. 35 (1987), 449-455. · Zbl 0647.46001
[5] BURZYK J.: Independence of sequences in convergence linear spaces. General Topology and its Relations to Modern Analysis and Algebra VI, Proc. of the Sixth Prague Topological Symposium 1986, Heldermann Verlag, Berlin, 1988, pp. 49-59.
[6] BURZYK J.: Decompositions of F-spaces into spaces with properties K, N, or \kappa . Generalized Functions and Convergence, Memorial Volume for Professor Jan Mikusiňski, World Scientific, Singapore, 1990, pp. 317-329.
[7] BURZYK J.-KAMIŇSKI A.: Operations on convergences. Tatra Mt. Math. Publ. 14 (1998), 199-212. · Zbl 0938.54005
[8] BURZYK J.-KLIS C.-LIPECKI Z.: On metrizable Abelian groups with a completeness-type property. Colloq. Math. 49 (1984), 33-39. · Zbl 0552.46001
[9] CHERESIZ V. M.: On the weak completeness of the dual to a convergence space. Dokl. Akad. Nauk SSSR 201 (1971), 548-551.
[10] DREWNOWSKI L.: A solution to a problem of De Wilde and Tsirulnikov. Manuscripta Math. 37 (1982), 61-64. · Zbl 0486.46003
[11] DREWNOWSKI L.-LABUDA I.-LIPECKI Z.: Existence of quasi-bases for separable topological linear spaces. Arch. Math. (Basel) 37 (1981), 454-456. · Zbl 0491.46005
[12] DREWNOWSKI L.-LIPECKI Z.: On some dense subspaces of topological linear spaces. II. Comment. Math. Prace Mat. 28 (1989), 175-188. · Zbl 0770.46001
[13] FOGED L.: The Baire category theorem for Fréchet groups in which every null sequence has a summable subsequence. Topology Proc. 8 (1983), 259-266. · Zbl 0557.22002
[14] KLIS, C: An example of noncomplete normed (K)-space. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), 415-420. · Zbl 0393.46017
[15] LABUDA I.-LIPECKI Z.: On subseries convergent series and m-quasi-bases in topological linear spaces. Manuscripta Math. 38 (1982), 87-98. · Zbl 0496.46006
[16] LIPECKI Z.: On some dense subspaces of topological linear spaces. Studia Math. 77 (1984), 413-421. · Zbl 0552.46002
[17] MAZUR S.-ORLICZ W.: Sur les espaces metriques lineaires (II). Studia Math. 13 (1953), 137-179. · Zbl 0052.11103
[18] ROLEWICZ S.: Metric Linear Spaces. (2nd, PWN-Reidel, Warszawa-Dordrecht, 1984. · Zbl 0226.46001
[19] SOBOLEV S. L.: Introduction to the Theory of Cubature Formulae. Nauka, Moscow, 1974.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.