Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T07:43:06.145Z Has data issue: false hasContentIssue false
  • English
  • Français

Slicings, Selections and Their Applications

Published online by Cambridge University Press:  20 November 2018

N. Ghoussoub ,
B. Maurey  and
W. Schachermayer
N. Ghoussoub
Affiliation:
Department of Mathematics University of Bristish Columbia Vancouver, B.C. V6TIY4
B. Maurey
Affiliation:
U.F.R. de Mathématiques Université ParisVII 75251 Paris Cedex 05 France
W. Schachermayer
Affiliation:
Institut für Mathematik J. Kepler UniversitätA-4040 Linz Austria
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the past few years, much progress have been made on several open problems in infinite dimensional Banach space theory. Here are some of the most recent results:

  1. 1) The existence of boundedly complete basic sequences in a large class of Banach spaces including the ones with the so-called Radon-Nikodym property ([G-M2], [G-M4]).

  2. 2) The embedding of separable reflexive Banach spaces into reflexive spaces with basis (fZl).

  3. 3) The existence of long sequences of projections and hence of locally uniformly convex norms in the duals of Asplund spaces. ([F-G])

Keywords

46A5546B2049A27
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 3 , 01 June 1992 , pp. 483 - 504
Copyright
Copyright © Canadian Mathematical Society 1992

References

[B] Bourgin, R.D., Geometric aspects of convex sets with the Radon-Nikodym property, Lecture notes in Mathematics 993, Springer- Verlag, 1983.Google Scholar
[B-P] Borwein, J. and Preiss, D., A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), 517–527.Google Scholar
[D-F-J-P] Davis, W.J., Figiel, T., Johnson, W.B. and Pelczynski, A.. Factoring weakly compact operators, J. Funct. Analysis 17 (1974), 311–327.Google Scholar
[D] Debs, G., Sélections maximales d'une multi-application, Séminaire d'initiation à l'analyse. Choquet Rogalski-St. Raymond, 1978.1979.Google Scholar
[E] Edgar, G.A., Complex martingale convergence, Springer-Verlag, Lecture Notes 1116 (1985), 38–59.Google Scholar
[E-W] Edgar, G.A. and Wheeler, R.F., Topological properties of Banach spaces, Pac. J. Math. 115 (1984).Google Scholar
[Ek] Ekeland, I., Non convex minimization problems, Bull. A.M.S. 1 (1979), 443–474.Google Scholar
[F-G] Fabian, M. and Godefroy, G., The dual of every Asplund space admits a projectional resolution of the identity, Studia Math. (2) 91 (1989).Google Scholar
[G-Ml] Ghoussouband, N. Maurey, B., Gδ -embeddings in Hubert space,]. Funct. Anal. (1) 61 (1985), 72–97.Google Scholar
[G-M2] Ghoussouband, N. Maurey, B., Gδ -embeddings in Hilbert space II, J. Funct. Anal. (2) 78 (1988), 271–305.Google Scholar
[G-M3] Ghoussouband, N. Maurey, B., Hδ -embeddings in Hilbert space and Optimization on Gδ -sets, Memoirs of the A.M.S. 349 (1986).Google Scholar
[G-M4] Ghoussouband, N. Maurey, B., A non-linear method for constructing certain basic sequences in Banach spaces, Illinois J. Math. (3) 34 (1990), 607–613.Google Scholar
[G-M5] Ghoussouband, N. Maurey, B., Plurisubharmonic martingales and barriers in complex quasi-Banach spaces, Ann. Inst. Fourier (4) 39 (1989), 1007–1060.Google Scholar
[J-R] Jayne, J. and Rogers, A., Borel selectors for upper semi-continuous set valued maps, Acta. Math. 155 (1985),41–79.Google Scholar
[K-M] Kuratowski, K. and Mostowski, A., Set Theory with an introduction to descriptive set theory, Studies in logic and the foundation of mathematics. North-Holland 86 (1976).Google Scholar
[P] Phelps, R.R., Convex functions, monotone operators and differentiability, 1364. Lecture notes in Math., Springer-Verlag, 1989.Google Scholar
[Z] Zippin, M., Banach spaces with separable duals, Israel J. Math., (1988).Google Scholar