The Radon-Nikodym property and dentable sets in Banach spaces
HTML articles powered by AMS MathViewer
- by W. J. Davis and R. R. Phelps PDF
- Proc. Amer. Math. Soc. 45 (1974), 119-122 Request permission
Abstract:
In order to prove a Radon-Nikodym theorem for the Bochner integral, Rieffel [5] introduced the class of “dentable” subsets of Banach spaces. Maynard [3] later introduced the strictly larger class of “$s$-dentable” sets, and extended Rieffel’s result to show that a Banach space has the Radon-Nikodym property if and only if every bounded nonempty subset of $E$ is $s$-dentable. He left open, however, the question as to whether, in a space with the Radon-Nikodym property, every bounded nonempty set is dentable. In the present note we give an elementary construction which shows this question has an affirmative answer.References
- C. Bessaga and A. Pełczyński, On extreme points in separable conjugate spaces, Israel J. Math. 4 (1966), 262–264. MR 211244, DOI 10.1007/BF02771641
- V. L. Klee Jr., Some characterizations of reflexivity, Rev. Ci. (Lima) 52 (1950), no. nos. 3-4, 15–23. MR 43364
- Hugh B. Maynard, A geometrical characterization of Banach spaces with the Radon-Nikodym property, Trans. Amer. Math. Soc. 185 (1973), 493–500. MR 385521, DOI 10.1090/S0002-9947-1973-0385521-0
- I. Namioka, Neighborhoods of extreme points, Israel J. Math. 5 (1967), 145–152. MR 221271, DOI 10.1007/BF02771100
- M. A. Rieffel, Dentable subsets of Banach spaces, with application to a Radon-Nikodým theorem, Functional Analysis (Proc. Conf., Irvine, Calif., 1966) Academic Press, London; Thompson Book Co., Washington, D.C., 1967, pp. 71–77. MR 0222618
- R. E. Huff, Dentability and the Radon-Nikodým property, Duke Math. J. 41 (1974), 111–114. MR 341033
- R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis 17 (1974), 78–90. MR 0352941, DOI 10.1016/0022-1236(74)90005-6
- R. E. Huff and P. D. Morris, Dual spaces with the Krein-Milman property have the Radon-Nikodým property, Proc. Amer. Math. Soc. 49 (1975), 104–108. MR 361775, DOI 10.1090/S0002-9939-1975-0361775-9 G. A. Edgar, A noncompact Choquet theorem.
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 119-122
- MSC: Primary 46B05
- DOI: https://doi.org/10.1090/S0002-9939-1974-0344852-7
- MathSciNet review: 0344852