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Quasi-suprabarrelled spaces

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  09 April 2009

J. C. Ferrando  and
M. López-Pellicer
J. C. Ferrando
Affiliation:
Departmento de Matemáticas (ETSIA)Universidad PolitécnicaApartado 22012 46022-Valencia, Spain
M. López-Pellicer
Affiliation:
Departmento de Matemáticas (ETSIA)Universidad PolitécnicaApartado 22012 46022-Valencia, Spain
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Abstract

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In this paper a proper class of barrelled spaces which strictly contains the suprabarrelled spaces is considered. A closed graph theorem and some permanence properties are given. This allows us to prove the necessity of a condition of a theorem of S. A. Saxon and P. P. Narayanaswami by constructing an example of a non-suprabarrelled Baire-like space which is a dense subspace of a Fréchet space and is not an (LF)-space under any strong locally convex topology.

Keywords

barrelled space.

MSC classification

Secondary: 46A30: Open mapping and closed graph theorems; completeness (including $B$-, $B_r$-completeness)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 1 , February 1989 , pp. 137 - 145
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]DeWilde, M., Closed graph theorems and webbed spaces (Pitman, London, San Francisco, Melbourne, 1978).Google Scholar
[2]Grothendieck, A., Topological vector spaces (Gordon and Breach, New York, London, Paris, 1973).Google Scholar
[3]Köthe, G., Topological vector spaces I (Springer-Verlag, Berlin, Heidelberg, New York, 1969).Google Scholar
[4]Narayanaswami, P. P. and Saxon, S. A., ‘(LF)-spaces, quasi-Baire spaces and the strongest locally convex topology’, Math. Ann. 274 (1986), 627–641.CrossRefGoogle Scholar
[5]Pérez, P. and Bonet, J., ‘Remarks and examples concerning suprabarrelled and totally barrelled spaces’, Arch. Math. 39 (1982), 340–347.Google Scholar
[6]Robertson, W., Tweddle, I. and Yeomans, F., ‘On the stability of barrelled topologies III’, Bull. Austral. Math. Soc. 22 (1980), 99–112.CrossRefGoogle Scholar
[7]Saxon, S. A., ‘Nuclear and product spaces, Baire-like spaces and the strongest locally convex topology’, Math. Ann. 197 (1972), 91–96.CrossRefGoogle Scholar
[8]Saxon, S. A. and Levin, M., ‘Every countable-codimensional subspace of a barrelled space is barrelled’, Proc. Amer. Math. Soc. 29 (1971), 91–96.CrossRefGoogle Scholar
[9]Saxon, S. A. and Narayanaswami, P. P., ‘Metrizable (LF)-spaces, (db)-spaces and the separable quotient problem’, Bull. Austral. Math. Soc. 23 (1981), 65–80.CrossRefGoogle Scholar
[10]Todd, A. and Saxon, S. A., ‘A property of locally convex Baire spaces’, Math. Ann. 206 (1973), 23–34.CrossRefGoogle Scholar
[11]Valdivia, M., ‘Absolutely convex sets in barrelled spaces’, Ann. Inst. Fourier 21 (1971), 3–13.CrossRefGoogle Scholar
[12]Valdivia, M., ‘On suprabarrelled spaces’, Functional Anal., Holomorphy and Approximation Theory, Rio de Janeiro 1978, pp. 572–580, Lecture Notes in Mathematics 843, Springer, Berlin, Heidelberg, New York, (1981).CrossRefGoogle Scholar
[13]Valdivia, M., ‘Śobre el teorema de la gráfica cerrada’, Collect. Math. 22 (1971), 51–72.Google Scholar
[14]Valdivia, M., ‘The space of distributions D'(Ω) is not Br-complete’, Math. Ann. 211 (1974), 145–149.CrossRefGoogle Scholar
[15]Valdivia, M., Topics in locally convex spaces (Math. Studies, North-Holland, Amsterdam, New York, Oxford, 1982).Google Scholar
[16]Valdivia, M. and Carreras, P. Pérez, ‘On totally barrelled spaces’, Math. Z. 1978 (1981), 263–269.CrossRefGoogle Scholar
[17]Valdivia, M. and Carreras, P. Pérez, ‘Sobre espacios (LF) metrizables’, Collect. Math. 33 (1982), 297–303.Google Scholar