Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Radon-Nikodym property and dentable sets in Banach spaces


Authors: W. J. Davis and R. R. Phelps
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] C. Bessaga and A. Pełczyński, On extreme points in separable conjugate spaces, Israel J. Math. 4 (1966), 262-264. MR 35 #2126. MR 0211244 (35:2126)
  • [2] V. L. Klee, Some characterizations of reflexivity, Rev. Ci. Lima 52 (1950), 15-23. MR 13, 250. MR 0043364 (13:250a)
  • [3] H. B. Maynard, A geometrical characterization of Banach spaces having the Radon-Nikodým property, Trans. Amer. Math. Soc. 185 (1973), 493-500. MR 0385521 (52:6382)
  • [4] I. Namioka, Neighborhoods of extreme points, Israel J. Math. 5 (1967), 145-152. MR 36 #4323. MR 0221271 (36:4323)
  • [5] 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 36 #5668. MR 0222618 (36:5668)
  • [6] R. E. Huff, Dentability and the Radon-Nikodym property, Duke Math. J. (to appear). MR 0341033 (49:5783)
  • [7] R. R. Phelps, Dentability and extreme points in Banach spaces, J. Functional Analysis (to appear). MR 0352941 (50:5427)
  • [8] R. E. Huff and P. D. Morris, Dual spaces with the Krein-Milman property have the Radon-Nikodým property, Proc. Amer. Math. Soc. (to appear). MR 0361775 (50:14220)
  • [9] G. A. Edgar, A noncompact Choquet theorem.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B05

Retrieve articles in all journals with MSC: 46B05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0344852-7
Keywords: Banach space, dentable sets, vector valued measures, Radon-Nikodym theorem
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society