Hostname: page-component-76c49bb84f-887v8 Total loading time: 0 Render date: 2025-07-12T18:44:26.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Weakly compactly generated locally convex spaces

Published online by Cambridge University Press:  24 October 2008

Richard J. Hunter  and
John Lloyd
Richard J. Hunter
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
John Lloyd
Affiliation:
Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

Locally convex spaces which are generated by a weakly compact subset (or a sequence of weakly compact subsets) and their subspaces are studied. Various characterizations and the permanence properties of these spaces are obtained. Certain results valid for weakly compactly generated Banach spaces are extended. These spaces are shown to have sequential properties which extend well-known properties of separable locally convex spaces.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 1 , July 1977 , pp. 85 - 98
Copyright
Copyright © Cambridge Philosophical Society 1977

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.)

Article purchase

Temporarily unavailable

References

REFERENCES

(1)Amemiya, I.Some examples of (F)- and (DF)-spaces. Proc. Japan Acad. 33 (1957), 169–171.Google Scholar
(2)Amir, D. and Lindenstrauss, J.The structure of weakly compact sets in Bach spaces. Ann. of Math. 88 (1968), 35–46.CrossRefGoogle Scholar
(3)Buchwalter, H.Espaces localement convexes semi-faibles. C. R. A. Sci. Paris A 278 (1974), 257–259.Google Scholar
(4)Bukvalov, A. V.On locally convex spaces generated by weakly compact sets. Veatnik Leningrad. Univ. Fiz. Him. 7 (1973), 11–17 (Russian).Google Scholar
(5)Corson, H. H. and Lindenstrauss, J.On weakly compact subsets of Bach spaces. Proc. Amer. Math. Soc. 17 (1966), 407–412.CrossRefGoogle Scholar
(6)Day, M.Strict convexity and smoothness of normed spaces. Trans. Amer. Math Soc. 78 (1955), 516–528.CrossRefGoogle Scholar
(7)Diestel, J.Geometry of Barach spaces – selected topics. Lecture Notes in Math. no. 485 (Berlin, Heidelberg, New York, Springer-Verlag, 1975).CrossRefGoogle Scholar
(8)Diestel, J., Morris, S. A. and Saxon, S. A.Varieties of locally convex topological linear spaces. Bull. Amer. Math. Soc. 77 (1971), 799–803.CrossRefGoogle Scholar
(9)Friedland, D.On closed subspaces of weakly compactly generated Bach spaces. Israel J. Math. (To appear.)Google Scholar
(10)Horváth, J.Topological vector spaces and distributions, vol. 1 (Reading, Addison-Wesley, 1966).Google Scholar
(11)John, K. and Zizler, V.Smoothness and its equivalents in weakly compactly generated Banach spaces. J. Functional Analysis 15 (1974), 1–11.CrossRefGoogle Scholar
(12)John, K. and Zizler, V.On the heredity of weak compact generating. Israel J. Math. 20 (1975), 228–236.Google Scholar
(13)Johnston, W. B. and Lindenstrauss, J.Some remarks on weakly compactly generated Banach spaces. Israel J. Math. 17 (1974), 219–230.CrossRefGoogle Scholar
(14)Köthe, G.Topological vector spaces, vol. 1 (Berlin, Heidelberg, New York, Springer-Verlag, 1969).Google Scholar
(15)Lindenstrauss, J.Weakly compact sets, their topological properties and the Banach spaces they generate. Proc. Int. Symp. Topology, Ann. Math. Stud. 69 (1972), 235–273.Google Scholar
(16)Reif, J.Some remarks on subspaces of weakly compactly generated Banach spaces. Comment. Math. Univ. Carolinae 16 (1975), 787–793.Google Scholar
(17)Rosenthal, H. P.The heredity problem for weakly compactly generated Banach spaces. Compositio. Math. 28 (1974), 83–111.Google Scholar
(18)Valdivia, M.Some criteria for weak compactness. J. Reine Angew. Math. 255 (1971), 165–169.Google Scholar
(19)Webb, J. H.Sequential convergence in locally convex spaces. Proc. Cambridge Philos. Soc. 64 (1968), 341–364.Google Scholar
(20)Zizler, V.Some corollaries of one Amir-Lindenstrauss's theorem on weakly compactly generated Banach spaces. Mathematics (Cluj). 15 (38) (1973), 331–333.Google Scholar
(21)Benyamini, Y., Rudin, M. E. and Wage, M. Continuous images of weakly compact subsets of Bach spaces. (Preprint.)Google Scholar