Local completeness and dual local quasi-completeness

Qiu, Jinghui

doi:10.1090/S0002-9939-00-05717-8

Local completeness and dual local quasi-completeness
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by Jinghui Qiu PDF

Proc. Amer. Math. Soc. 129 (2001), 1419-1425 Request permission

Abstract:

It is proved that $l^{q}$-completeness $(1<q<\infty )$ is equivalent to $l^{1}$-completeness (defined by Saxon and Sánchez Ruiz), and becomes a new characteristic condition for local completeness. The relationship between dual local completeness, dual local quasi-completeness and the Banach-Mackey property is investigated. For a quasi-Mackey space, dual local quasi-completeness, $c_{0}$-quasi-barrelledness, Ruess’ property (quasi-L) and $C$-quasi-barrelledness are equivalent to each other.

References

Marc De Wilde, Closed graph theorems and webbed spaces, Research Notes in Mathematics, vol. 19, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1978. MR 518869
P. Dierolf, Une characterisation des espaces vectoriels complets au sens de Mackey, C. R. Acad. Sci. Paris. 283 (1976), 245-248.
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. MR 880207
W. Ruess, A Grothendieck representation for the completion of cones of continuous seminorms, Math. Ann. 208 (1974), 71–90. MR 361702, DOI 10.1007/BF01432521
Stephen A. Saxon and L. M. Sánchez Ruiz, Dual local completeness, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1063–1070. MR 1389535, DOI 10.1090/S0002-9939-97-03864-1
Stephen A. Saxon and L. M. Sánchez Ruiz, Mackey weak barrelledness, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3279–3282. MR 1458260, DOI 10.1090/S0002-9939-98-04511-0
S. A. Saxon and L. M. Sánchez Ruiz, Reinventing weak barrelledness, preprint.
Bella Tsirulnikov, On conservation of barrelledness properties in locally convex spaces, Bull. Soc. Roy. Sci. Liège 49 (1980), no. 1-2, 5–25 (English, with French summary). MR 586948
Albert Wilansky, Modern methods in topological vector spaces, McGraw-Hill International Book Co., New York, 1978. MR 518316