Abstract
It is shown that for Banach spaces the Radon-Nikodym property and the Bishop-Phelps property are equivalent. Using similar techniques, we prove that ifC is a bounded, closed and convex subset of a Banach space such that every nonempty subset ofC is dentable, then the strongly exposing functionals ofC form a denseG δ-subset of the dual.
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References
E. Bishop and R. R. Phelps,The support functionals of a convex set, Proc. Symp. Pure Math.7, (Convexity) 27–35.
W. J. Davis and R. R. Phelps,The Radon-Nikodym propoerty and dentable sets in Banach spaces, Proc. Amer. Math. Soc.45 (1974), 119–122.
J. Diestel,Geometry of Banach Spaces—Selected Topics, Springer-Verlag 485, 1975.
R. E. Huff,Dentability and the Radon-Nikodym property, Duke Math. J.41 (1974), 111–114.
R. E. Huff and P. Morris,Geometric characterizations of the Radon-Nikodym property in Banach spaces.
J. Lindenstrauss,On operators which attain their norm, Israel J. Math.1 (1963), 139–148.
H. Maynard,A geometric characterization of Banach spaces possessing the Radon-Nikodym property, Trans. Amer. Math. Soc.185 (1973), 493–500.
R. R. Phelps,Dentability and extreme points in Banach spaces, J. Functional Analysis, (1974).
M. A. Rieffel,Dentable subsets of Banach spaces, with applications to a Radon-Nikodym theorem, Proc. Conf. Functional Analysis, Thompson Book Co., Washington, D.C., 1967, pp. 71–77.
S. L. Troyanski,On locally uniformely convex and differentiable norms in certain nonseparable Banach spaces, Studia Math.37 (1971), 173–180.
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Bourgain, J. On dentability and the Bishop-Phelps property. Israel J. Math. 28, 265–271 (1977). https://doi.org/10.1007/BF02760634
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DOI: https://doi.org/10.1007/BF02760634