Advertisement

Israel Journal of Mathematics

, Volume 37, Issue 1–2, pp 54–75 | Cite as

Martingales valued in certain subspaces ofL 1

  • J. Bourgain
  • H. P. Rosenthal
Article

Abstract

A Banach subspaceE ofL 1 is constructed such thatE fails the Radon-Nikodym property, yetE has no bounded dyadicδ-tree for anyδ>0. The unit ball ofE is also relatively compact in the topology of convergence in probability, which implies thatE has the strong Schur property. The construction uses martingales and probabilistic techniques.

Keywords

Unit Ball Independent Random Variable Difference Sequence Borel Subset Finite Dimensional Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Bourgain,Un espace non Radon-Nikodym sans arbre diadique, Seminaire d’Analyse Fonctionnelle 1978–79, Ecole Polytechnique, Palaiseau.Google Scholar
  2. 2.
    J. Bourgain,A non-dentable set without the RN property, Studia Math. (to appear).Google Scholar
  3. 3.
    J. Bourgain and H. P. Rosenthal,Geometrical implications of certain finite dimensional decompositions, to appear.Google Scholar
  4. 4.
    J. Diestel and J. J. Uhl, Jr.,Vector Measures, Amer. Math. Soc., Mathematical surveys # 15, Providence, 1977.zbMATHGoogle Scholar
  5. 5.
    W. B. Johnson and E. Odell,Subspaces of L p which embed into lp, Comp. Math.28 (1974), 37–49.zbMATHMathSciNetGoogle Scholar
  6. 6.
    J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces, Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 1973.zbMATHGoogle Scholar
  7. 7.
    M. Loève,Probability Theory, third edition, D. van Nostrand Company, Princeton, 1963.zbMATHGoogle Scholar
  8. 8.
    N. T. Peck,An L 0-compact convex set in L1 with no extreme points, preliminary report.Google Scholar
  9. 9.
    J. W. Roberts,Compact convex sets with no extreme points in the spacesLp ([0, 1]), 0≦p<1, to appear.Google Scholar
  10. 10.
    H. P. Rosenthal,A characterization of Banach spaces containing l 1, Proc. Nat. Acad. Sci. U.S.A.71 (1974), 2411–2413.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    H. P. Rosenthal,A subsequence splitting result for L 1 -bounded sequences of random variables, in preparation.Google Scholar
  12. 12.
    C. Stegall,The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc.206 (1975), 213–223.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    S. J. Szarek,On the best constant in the Khintchine inequality, Studia Math.63 (1976), 197–208.MathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1980

Authors and Affiliations

  • J. Bourgain
    • 1
    • 2
    • 3
  • H. P. Rosenthal
    • 1
    • 2
    • 3
  1. 1.University of Paris VIFrance
  2. 2.Free University of BrusselsBelgium
  3. 3.The University of Texas at AustinUSA

Personalised recommendations