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Local completeness and dual local quasi-completeness

Author: Jinghui Qiu
Journal: Proc. Amer. Math. Soc. 129 (2001), 1419-1425
MSC (2000): Primary 46A08
Published electronically: October 24, 2000
MathSciNet review: 1814168
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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 [Enhancements On Off] (What's this?)

  • [1] M. De Wilde, Closed Graph Theorems and Webbed Spaces, Pitman, 1978. MR 81j:46013
  • [2] P. Dierolf, Une characterisation des espaces vectoriels complets au sens de Mackey, C. R. Acad. Sci. Paris. 283 (1976), 245-248.
  • [3] G. Köthe, Topological Vector Spaces I, Springer-Verlag, 1969. MR 40:1750
  • [4] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, Math. Studies 131, North Holland, 1987. MR 88j:46003
  • [5] W. Ruess, A Grothendieck representation for the completion of cones of continuous seminorms, Math. Ann. 208(1974), 71-90. MR 50:14147
  • [6] S. A. Saxon and L. M. Sánchez Ruiz, Dual local completeness, Proc. Amer. Math. Soc. 125(1997), 1063-1070. MR 97g:46004
  • [7] S. A. Saxon and L. M. Sánchez Ruiz, Mackey weak barrelledness, Proc. Amer. Math. Soc. 126(1998), 3279-3282. MR 99a:46006
  • [8] S. A. Saxon and L. M. Sánchez Ruiz, Reinventing weak barrelledness, preprint.
  • [9] B. Tsirulnikov, On conservation of barrelledness properties in locally convex spaces, Bull. Soc. Royale Sci. Liège 49(1980), 5-25. MR 82c:46005
  • [10] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, 1978. MR 81d:46001

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Additional Information

Jinghui Qiu
Affiliation: Department of Mathematics, Suzhou University, Suzhou 215006, People’s Republic of China

Keywords: Locally complete, dual locally quasi-complete, Banach-Mackey property, quasi-Mackey spaces
Received by editor(s): September 28, 1998
Received by editor(s) in revised form: August 17, 1999
Published electronically: October 24, 2000
Communicated by: Dale E. Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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