zbMATH — the first resource for mathematics

Preserved extreme points. (English. Russian original) Zbl 0591.46004
Funct. Anal. Appl. 19, 144-146 (1985); translation from Funkts. Anal. Prilozh. 19, No. 2, 76-77 (1985).
An extreme point of the unit ball in a Banach space X is said to be preserved if its image under the canonical mapping from X into its second dual \(X^{**}\) is an extreme point of the unit ball in \(X^{**}\). The author proves that X is reflexive if and only if every extreme point of its unit ball is preserved in each equivalent norm.
Reviewer: C.M.Edwards
46A55 Convex sets in topological linear spaces; Choquet theory
46B10 Duality and reflexivity in normed linear and Banach spaces
Full Text: DOI
[1] R. R. Phelps, ”Extreme points of polar convex sets,” Proc. Am. Math. Soc.,12, No. 2, 291-296 (1961). · Zbl 0099.31603 · doi:10.1090/S0002-9939-1961-0121634-3
[2] J. Diestel and J. J. Uhl, Jr., Vector Measures, Mathematical Surveys, No. 15, Am. Math. Soc., Providence (1977).
[3] I. Singer, Bases in Banach Spaces I, Springer-Verlag, Berlin?New York (1970). · Zbl 0198.16601
[4] V. F. Gaposhkin and M. I. Kadets, Mat. Sb.,61, No. 1, 1-12 (1963).
[5] P. Morris, ”Disappearance of extreme points,” Proc. Am. Math. Soc.,88, No. 2, 244-246 (1983). · Zbl 0541.46013 · doi:10.1090/S0002-9939-1983-0695251-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.