Abstract
We give an exposition of the “hard case” of Bourgain's theorem, that a Banach space X has RNP iff each subspace with a finite dimensional decomposition has RNP. We reproduce essentially Bourgain's arguments, by explaining the ideas underlying the proof and giving slightly altered arguments for some of the technical details.
Keywords
- Banach Space
- Unit Sphere
- Finite Subset
- Finite Dimensional Subspace
- Martingale Difference
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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
S. Argyros, E. Odell, H.P. Rosenthal. In preparation
J. Bourgain. Dentability and finite-dimensional decompositions. Studia Math. 67 (1980), 135–148.
J. Bourgain. La propriété de Radon-Nikodym. Publications Mathématiques de l'Université Pierre et Marie Curie, no. 36, (1979).
J. Bourgain, H.P. Rosenthal. Geometrical implications of certain finite-dimensional decompositions. Bull. Soc. Math. Belg. 32 (1980), 57–82.
J. Diestel, J.J. Uhl. Vector measures. AMS, Providence, 1977.
G.A. Edgar, R. Wheeler. Topological properties of Banach spaces. Pac. J. of Math. 115 (1984), 317–350.
N. Ghoussoub, G. Godefroy, B. Maurey. First class functions around a subset of a compact space and geometrically regular Banach spaces. Preprint.
N. Ghoussoub, G. Godefroy, B. Maurey, W. Schachermayer. Some topological and geometrical structures in Banach spaces (essentially containing [G-G-M]), (1986), to appear.
N. Ghoussoub, B. Maurey, W. Schachermayer. A counter-example to a problem about points of continuity in Banach spaces. (1985), to appear in Proc. of AMS.
N. Ghoussoub, B. Maurey, W. Schachermayer. Geometrical implications of certain infinitedimensional decompositions. In preparation.
R.C. James. Subbushes and extreme points in Banach spaces. In “Proceedings of a Research Workshop in Bach Spaces Theory”, University of Iowa, editor: Bor-Luh-Lin, (1981), 59–81.
J. Lindenstrauss, L. Tzafriri. Classical Banach Spaces I. Springer (1977).
W. Schachermayer. The Radon-Nikodym property and the Krein-Milman property are equivalent for strongly regular sets. (1986), to appear in Transactions of the AMS.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag
About this paper
Cite this paper
Schachermayer, W. (1987). On a theorem of J. Bourgain on finite dimensional decompositions and the radon-nikodym property. In: Lindenstrauss, J., Milman, V.D. (eds) Geometrical Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1267. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078139
Download citation
DOI: https://doi.org/10.1007/BFb0078139
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-18103-3
Online ISBN: 978-3-540-47771-6
eBook Packages: Springer Book Archive
