Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. M. Day,Normed Linear Spaces, Academic Press, New York, 1962.
R. C. James,Bases and reflexivity of Banach spaces, Bull. Amer. Math Soc.56 (1950), 58 (abstract 80).
R. C. James,Characterizations of reflexivity, Studia Math.23 (1964), 205–216.
R. C. James,Weakly compact sets, Trans. Amer. Math. Soc.113 (1964), 129–140.
J. L. Kelley and I. Namioka,Linear Topological Spaces, D. Van Nostrand, Princeton, 1963.
V. Klee,Some characterizations of reflexity, Rev. Ci. (Lima)52 (1950), 15–23.
J. D. Pryce,Weak compactness in locally convex spaces, Proc. Amer. Math. Soc.17 (1966), 148–155.
H. H. Schaefer,Topological Vector Spaces, Macmillan, New York, 1966.
S. Simons,A convergence theorem with boundary, Pacific J. Math40 (1972), 703–708.
S. Simons,Maximinimax, minimax, and antiminimax theorems and a result of R. C. James, Pacific J. Math.40 (1972), 709–718.
Author information
Authors and Affiliations
Additional information
This work was supported in part by National Science Foundation grant GP-28578.
Rights and permissions
About this article
Cite this article
James, R.C. Reflexivity and the sup of linear functionals. Israel J. Math. 13, 289–300 (1972). https://doi.org/10.1007/BF02762803
Issue Date:
DOI: https://doi.org/10.1007/BF02762803