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

Published online by Cambridge University Press:  18 May 2009

Lech Drewnowski ,
Miguel Florencio  and
Pedro J. Paúl
Lech Drewnowski
Affiliation:
Instytut Matematyki, Uniwertsytet A. Mickiewicza, Matejki 48-49, 60-769 Poznań, Poland Department of Mathematics, University of Mississippi, University MS38677, U.S.A. E-mail: DREWLECH@PLPUAM11.BITNET
Miguel Florencio
Affiliation:
E.S. Ingenieros Industriales, Avda Reina Mercedes S/N, 41012-Sevilla, Spain E-mail: PITI%ETSII.US.ES@CERNVAX and PITI%CICA.ES@CERNVAX
Pedro J. Paúl
Affiliation:
E.S. Ingenieros Industriales, Avda Reina Mercedes S/N, 41012-Sevilla, Spain E-mail: PITI%ETSII.US.ES@CERNVAX and PITI%CICA.ES@CERNVAX
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 1 , January 1994 , pp. 57 - 69
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Bennett, G., Some inclusion theorems for sequence spaces, Pacific J. Math. 46 (1973), 17–30.CrossRefGoogle Scholar
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.Bennett, G. and Kalton, N. J., Inclusion theorem for K-spaces, Canad. J. Math. 25 (1973), 511–524.CrossRefGoogle Scholar
4.Buck, R. C., The measure theoretic approach to density, Amer. J. Math. 68 (1946), 560–580.CrossRefGoogle Scholar
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.Diestel, J. and Uhl, J. J. Jr, Vector measures, Mathematical Surveys No. 15, (American Mathematical Society, Providence, RI, 1977).CrossRefGoogle Scholar
7.Estrada, R. and Kanwal, R. P., Series that converge on sets of null density, Proc. Amer. Math. Soc. 97 (1986), 682–686.CrossRefGoogle Scholar
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.Freniche, F. J., The Vitali–Hahn–Saks theorem for Boolean algebras with the subsequential interpolation property, Proc. Amer. Math. Soc. 92 (1984), 362–366.CrossRefGoogle Scholar
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.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.Köthe, G., Topological vector spaces I, (Springer-Verlag, 1969).Google Scholar
13.Noll, D. and Stadler, W., Abstract sliding hump technique and characterization of barrelled spaces, Studia Math. 94 (1989), 103–120.CrossRefGoogle Scholar
14.Carreras, P. Péerez and Bonet, J., Barrelled locally convex spaces, Notas de Matemáatica No. 131, (North-Holland, 1987).CrossRefGoogle Scholar
15.Ruckle, W. H., Sequence spaces, Research Notes in Mathematics, Vol. 49, (Pitman, 1981).Google Scholar
16.Schachermayer, W., On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissert. Math. 214 (1982), 1–33.Google Scholar
17.Singer, I., Bases in Banach spaces, II (Springer-Verlag, 1981).CrossRefGoogle Scholar
18.Swetits, J., A characterization of a class of barrelled sequence spaces, Glasgow Math. J. 19 (1978), 27–31.CrossRefGoogle Scholar
19.Webb, J. H., Sequential convergence in locally convex spaces, Proc. Cambridge Phil. Soc. 64 (1968), 341–364.CrossRefGoogle Scholar
20.Wilansky, A., Modern methods in topological vector spaces (McGraw-Hill, New York, 1978).Google Scholar