Abstract

In his study of the Radon-Nikodym property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set A that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set A (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of A-valued quasimartingales with sharply divergent behavior.

Author information

Contact details are reproduced from the original publication and may be historical.

Stephen J. Dilworth

Dept. of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

dilworth@math.sc.edu

Chris Gartland

Dept. of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.

cgartla2@illinois.edu

Denka Kutzarova

Dept. of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.
and: Inst. of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

denka@math.uiuc.edu

N. Lovasoa Randrianarivony

Dept. of Mathematics and Statistics, Saint Louis University, St. Louis, MO 63103, U.S.A.

nrandria@slu.edu

Keywords

  • Dentable sets in normed spaces
  • martingale convergence
  • Radon-Nikodým property
  • convex sets
  • extreme points

Mathematics Subject Classification

46B22; 46B20, 52A07, 60G42

References

  1. J. Bourgain: La Propriété de Radon-Nikodým: Cours de 3ième Cycle, Université Pierre et Marie Curie, Paris VI (1979).Google Scholar
  2. N. Dunford, J. T. Schwartz: Linear Operators. I: General Theory, with the Assistance of W. G. Bade and R. G. Bartle, Pure and Applied Mathematics 7, Interscience Publishers, New York (1958).Google Scholar
  3. H. B. Maynard: A geometrical characterization of Banach spaces with the Radon-Nikodým property, Trans. Amer. Math. Soc. 185 (1973) 493–500.https://doi.org/10.1090/S0002-9947-1973-0385521-0Google Scholar

Suggested citation

S. J. Dilworth, C. Gartland, D. Kutzarova, N. L. Randrianarivony. “Nondentable Sets in Banach Spaces.” Journal of Convex Analysis 28 (2021), No. 1, 31–40. https://doi.org/10.68381/jca28004

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.