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.


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  1. 1.
    M. M. Day,Normed Linear Spaces, Academic Press, New York, 1962.zbMATHGoogle 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.zbMATHMathSciNetGoogle Scholar
  4. 4.
    R. C. James,Weakly compact sets, Trans. Amer. Math. Soc.113 (1964), 129–140.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. L. Kelley and I. Namioka,Linear Topological Spaces, D. Van Nostrand, Princeton, 1963.zbMATHGoogle 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.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    H. H. Schaefer,Topological Vector Spaces, Macmillan, New York, 1966.zbMATHGoogle Scholar
  9. 9.
    S. Simons,A convergence theorem with boundary, Pacific J. Math40 (1972), 703–708.zbMATHMathSciNetGoogle Scholar
  10. 10.
    S. Simons,Maximinimax, minimax, and antiminimax theorems and a result of R. C. James, Pacific J. Math.40 (1972), 709–718.zbMATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1972

Authors and Affiliations

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

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