Israel Journal of Mathematics

, Volume 13, Issue 3–4, pp 289–300 | Cite as

Reflexivity and the sup of linear functionals

  • Robert C. James


A relatively easy proof is given for the known theorem that a Banach space is reflexive if and only if each continuous linear functional attains its sup on the unit ball. This proof simplifies considerably for separable spaces and can be extended to give a proof that a boundedw-closed subsetX of a complete locally convex linear topological space isw-compact if and only if each continuous linear functional attains its sup onX.


Banach Space Unit Ball Topological Vector Space Linear Functional Weak Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. M. Day,Normed Linear Spaces, Academic Press, New York, 1962.MATHGoogle Scholar
  2. 2.
    R. C. James,Bases and reflexivity of Banach spaces, Bull. Amer. Math Soc.56 (1950), 58 (abstract 80).Google Scholar
  3. 3.
    R. C. James,Characterizations of reflexivity, Studia Math.23 (1964), 205–216.MATHMathSciNetGoogle Scholar
  4. 4.
    R. C. James,Weakly compact sets, Trans. Amer. Math. Soc.113 (1964), 129–140.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. L. Kelley and I. Namioka,Linear Topological Spaces, D. Van Nostrand, Princeton, 1963.MATHGoogle Scholar
  6. 6.
    V. Klee,Some characterizations of reflexity, Rev. Ci. (Lima)52 (1950), 15–23.MathSciNetGoogle Scholar
  7. 7.
    J. D. Pryce,Weak compactness in locally convex spaces, Proc. Amer. Math. Soc.17 (1966), 148–155.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    H. H. Schaefer,Topological Vector Spaces, Macmillan, New York, 1966.MATHGoogle Scholar
  9. 9.
    S. Simons,A convergence theorem with boundary, Pacific J. Math40 (1972), 703–708.MATHMathSciNetGoogle Scholar
  10. 10.
    S. Simons,Maximinimax, minimax, and antiminimax theorems and a result of R. C. James, Pacific J. Math.40 (1972), 709–718.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1972

Authors and Affiliations

  • Robert C. James
    • 1
  1. 1.Claremont Graduate SchoolClaremont

Personalised recommendations