Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T13:14:38.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Sequential convergence in locally convex spaces

Published online by Cambridge University Press:  24 October 2008

J. H. Webb
J. H. Webb
Affiliation:
Gonville and Caius College, Cambridge and University of Cape Town, South Africa
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a locally convex Hausdorff linear topological space, we construct and examine the following topologies in the space:

the finest locally convex topology with the same convergent sequences as the initial topology, and the finest locally convex topology with the same precompact sets as the initial topology.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 2 , April 1968 , pp. 341 - 364
Copyright
Copyright © Cambridge Philosophical Society 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Dieudonné, J.Sur les espaces de Köthe. J. Analyse Math. 1 (1951), 81–115.CrossRefGoogle Scholar
(2)Dieudonné, J. and Gomes, A. P.Sur certains espaces vectoriels topologiques. C.R. Acad. Sci. Paris 230 (1950), 1129–1130.Google Scholar
(3)Dudley, R. M.On sequential convergence. Trans. Amer. Math. Soc. 112 (1964), 483–507.CrossRefGoogle Scholar
(4)Garling, D. J. H.A generalized form of inductive-limit topology for vector spaces. Proc. London Math. Soc. 14 (1964), 1–28.CrossRefGoogle Scholar
(5)Garling, D. J. H.Weak Cauchy sequences in normed linear spaces. Proc. Cambridge Philos. Soc. 60 (1964), 817–819.CrossRefGoogle Scholar
(6)Green, H. F.Convergence in sequence spaces. Proc. Edinburgh Math. Soc. 11 (1958/1959), 83–85.CrossRefGoogle Scholar
(7)Grothendieck, A.Sur les espaces (F) et (DF). Summa Brasil Math. 3 (1954), 57–123.Google Scholar
(8)Grothendieck, A.Espaces vectoriels topologiques (3rd edition; São Paulo, 1964).Google Scholar
(9)Kantorovich, L. V. and Akilov, G. P.Functional analysis in normed spaces (Pergamon, 1964).Google Scholar
(10)Kelley, J. L., Namioka, I. and co-authors. Linear topological spaces (Van Nostrand, 1963).CrossRefGoogle Scholar
(11)Kisyński, J.Convergence du type L. Colloq. Math. 7 (1960), 205–211.CrossRefGoogle Scholar
(12)Köthe, G.Neubegründung der Theorie der vollkommenen Räume. Math. Nachr. 4 (1951), 70–80.CrossRefGoogle Scholar
(13)Köthe, G.Topologische lineare Räume(Springer, 1960).CrossRefGoogle Scholar
(14)Levin, V. L.O klasse lokal'no vypuklykh prostranstv. Dokl. Akad. Nauk SSSR 145 (1962), 35–37. (Translation: On a class of locally convex spaces. Soviet Math. 3 (1962), 929–931).Google Scholar
(15)Shirai, T.Sur les topologies des espaces de L. Schwartz. Proc. Jap. Acad. 35 (1959), 31–36.Google Scholar
(16)Welland, R.On Köthe spaces. Trans. Amer. Math. Soc. 112 (1964), 267–277.Google Scholar
(17)Yoshinaga, K.On a locally convex space introduced by J. S. e Silva. J. Sci. Hiroshima Univ. Ser. A-I Math. 21 (1957), 89–98.Google Scholar