Skip to main content
SpringerLink
Log in

Weak compactness and reflexivity

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Several characterizations of weak-compactness are given for subsets of complete locally convex linear topological spaces and of Banach spaces. Some are new and some are generalizations of known facts.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Banach,Thèorie des opérations linéaires, Warsaw, 1932.

  2. M. M. Day,Normed linear spaces, Academic Press, New York, 1962.

    MATH  Google Scholar 

  3. W. F. Eberlein,Weak compactness in Banach spaces, Proc. Nat. Acad. Sci. U.S.A.33 (1947), 51–53.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Eidelheit,Zur Theorie der konvexen Mengen in linearten normierten Räumen, Studia Math.6 (1963), 104–111.

    Google Scholar 

  5. V. Gantmaher and V. Smul’yan,Sur les espaces linéaires dont la sphère unitaire est faiblement compacte, Dokl. Akad. Nauk SSSR17 (1937), 91–94.

    Google Scholar 

  6. A. Grothendieck,Critères de compacticité dans les espaces fonctionnels généraux, Amer. J. Math.74 (1952), 168–186.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. M. Grunblum,Certain théorèmes sur la base dans un espace de type (B), Dokl. Akad. Nauk SSSR,31 (1941), 428–432.

    MathSciNet  Google Scholar 

  8. R. C. James,Reflexivity and the supremum of linear functionals, Ann. of Math.66 (1957), 159–169.

    Article  MathSciNet  Google Scholar 

  9. R. C. James,Characterizations of reflexivity, Studia Math.23 (1964), 205–216.

    MATH  MathSciNet  Google Scholar 

  10. R. C. James,Weak compactness and separation, Canad. J. Math.16 (1964), 204–206.

    MATH  MathSciNet  Google Scholar 

  11. R. C. James,Weakly compact sets, Trans. Amer. Math. Soc.113 (1964), 129–140.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. L. Kelley and I. Namioka,Linear topological spaces, Van Nostrand, Princeton, 1963.

    MATH  Google Scholar 

  13. V. L. Klee, Jr.,Convex sets in linear spaces, Duke Math. J.18 (1951), 875–883.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Krein and V. Smul’yan,On regularly convex sets in the space conjugate to a Banach space, Ann. of Math.41 (1940), 556–583.

    Article  MathSciNet  Google Scholar 

  15. D. Milman,On some criteria for the regularity of spaces of the type (B), Dokl. Akad, Nauk SSSR20 (1938), 243–246.

    Google Scholar 

  16. D. P. Milman and V. D. Milman,Some geometric properties of nonreflexive spaces, Soviet Math.4 (1963), 1250–1252.

    Google Scholar 

  17. A. Pełczyński,A note on the paper of I. Singer “Basic sequences and reflexivity of Banach spaces”, Studia Math.21 (1962), 371–374.

    MathSciNet  Google Scholar 

  18. V. Pták,Two remarks on weak compactness, Czechoslovak Math. J.5 (1955), 532–545.

    MathSciNet  Google Scholar 

  19. V. Pták,Biorthogonal systems and reflexivity of Banach spaces, Czechoslovak Math. J.9 (1959), 319–326.

    MathSciNet  Google Scholar 

  20. V. Smul’yan,On the principle of inclusion in the space of type (B), Mat. Sb.5 (1939), 317–328.

    MATH  Google Scholar 

  21. J. W. Tukey,Some notes on the separation of convex sets, Portugal. Math.3 (1942), 95–102.

    MATH  MathSciNet  Google Scholar 

  22. K. Pełczyński,A proof of Eberlein-Šmulian theorem by an application of basic sequences, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.12 (1964), 539–544.

    Google Scholar 

  23. I. Singer,Basic sequences and reflexivity of Banach spaces, Studia Math.21 (1962), 351–369.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Harvey Mudd College, Claremont, California

    Robert C. James

Authors
  1. Robert C. James
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Lecture delivered at a symposium on Series and Geometry in Linear Spaces, held at the Hebrew University of Jerusalem from March 16, till March 24, 1964.

Rights and permissions

Reprints and permissions

About this article

Cite this article

James, R.C. Weak compactness and reflexivity. Israel J. Math. 2, 101–119 (1964). https://doi.org/10.1007/BF02759950

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02759950

Keywords

Search

Navigation