Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections

Drewnowski, Lech ; Florencio, Miguel ; Paúl, Pedro J.

Glasgow mathematical journal, Tome 36 (1994) no. 1, pp. 57-69

Résumé

The research presented in this paper started by extending a theorem of Swetits [18]about barrelledness of subspaces of metrizable AK-spaces to general AK-spaces of scalar sequences. The extension reads as follows.(1) A subspace λ0 of a barrelled AK-space λ such that λ0 ⊃ φ is barrelled if and only if its dualis weak* sequentially complete. If in addition λ0 is monotone, then it is barrelled if and only ifequals the Köthe dualof λ0.As an easy consequence of this extension, we obtained the following result of Elstrodt and Roelcke [8, Corollary 3.4].(2) If λ is a barrelled monotone AK-space, then also its subspace L(λ), consisting of all sequences in λ with zero-density support, is barrelled.

Détail

DOI : 10.1017/S0017089500030548

Bibliographie

[1] 1.Bennett, G., Some inclusion theorems for sequence spaces, Pacific J. Math. 46 (1973), 17–30. Google Scholar | DOI

[2] 2.Bennett, G., A new class of sequence spaces with applications in summability theory, J.Reine Angew. Math. 266 (1974), 49–75. Google Scholar

[3] 3.Bennett, G. and Kalton, N. J., Inclusion theorem for K-spaces, Canad. J. Math. 25 (1973), 511–524. Google Scholar | DOI

[4] 4.Buck, R. C., The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560–580. Google Scholar | DOI

[5] 5.Dashiell, F. K., Non-weakly compact operators from order-Cauchy complete C(S) lattices, with applications to Baire classes, Trans. Amer. Math. Soc. 266 (1981), 397–413. Google Scholar

[6] 6.Diestel, J. and Uhl, J. J. Jr, Vector measures, Mathematical Surveys No. 15, (American Mathematical Society, Providence, RI, 1977). Google Scholar | DOI

[7] 7.Estrada, R. and Kanwal, R. P., Series that converge on sets of null density, Proc. Amer. Math. Soc. 97 (1986), 682–686. Google Scholar | DOI

[8] 8.Elstrodt, J. and Roelcke, W., Some dense barrelled subspaces of barrelled spaces with decomposition properties, Note Mat. 6 (1986), 155–203. Google Scholar

[9] 9.Freniche, F. J., The Vitali–Hahn–Saks theorem for Boolean algebras with the subsequential interpolation property, Proc. Amer. Math. Soc. 92 (1984), 362–366. Google Scholar | DOI

[10] 10.Freniche, F. J., Some classes of Boolean algebras related to the Vitali–Hahn–Saks and Nikodym property, Unpublished paper presented at a meeting in Oberwolfach in January, 1985. Google Scholar

[11] 11.Kamthan, P. K. and Gupta, M., Sequence spaces and series, Lecture Notes in Pure and Applied Mathematics, Vol. 65, (Marcel Dekker, Inc., New York and Basel, 1981). Google Scholar

[12] 12.Köthe, G., Topological vector spaces I, (Springer-Verlag, 1969). Google Scholar

[13] 13.Noll, D. and Stadler, W., Abstract sliding hump technique and characterization of barrelled spaces, Studia Math. 94 (1989), 103–120. Google Scholar | DOI

[14] 14.Carreras, P. Péerez and Bonet, J., Barrelled locally convex spaces, Notas de Matemáatica No. 131, (North-Holland, 1987). Google Scholar | DOI

[15] 15.Ruckle, W. H., Sequence spaces, Research Notes in Mathematics, Vol. 49, (Pitman, 1981). Google Scholar

[16] 16.Schachermayer, W., On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissert. Math. 214 (1982), 1–33. Google Scholar

[17] 17.Singer, I., Bases in Banach spaces, II (Springer-Verlag, 1981). Google Scholar | DOI

[18] 18.Swetits, J., A characterization of a class of barrelled sequence spaces, Glasgow Math. J. 19 (1978), 27–31. Google Scholar | DOI

[19] 19.Webb, J. H., Sequential convergence in locally convex spaces, Proc. Cambridge Phil. Soc. 64 (1968), 341–364. Google Scholar | DOI

[20] 20.Wilansky, A., Modern methods in topological vector spaces (McGraw-Hill, New York, 1978). Google Scholar