Hostname: page-component-7bb8b95d7b-fmk2r Total loading time: 0 Render date: 2024-10-06T22:44:53.575Z Has data issue: false hasContentIssue false
  • English
  • Français

The weak Banach-Saks property on Lp(μ, E)

Published online by Cambridge University Press:  24 October 2008

Pilar Cembranos
Pilar Cembranos
Affiliation:
Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Extract

A Banach space E is said to have the Banach-Saks property (BS) if every bounded sequence (xn) in E has a subsequence (x′n) with norm convergent Cesaro means; that is, there is x in E such that

If this occurs for every weakly convergent sequence in E it is said that E has the Weak Banach-Saks property (WBS) (also called Banach-Saks-Rosenthal property).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 115 , Issue 2 , March 1994 , pp. 283 - 290
Copyright
Copyright © Cambridge Philosophical Society 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aldous, D. J.. Unconditional bases and martingales in Lp(F). Math. Proc. Cambridge Phil. Soc. 85 (1979), 117–123.CrossRefGoogle Scholar
[2]Banach, S. and Saks, S.. Sur la convergence forte dans les champs Lp. Studia Math. 2 (1930), 51–57.CrossRefGoogle Scholar
[3]Beauzamy, B.. Banach–Saks properties and spreading models. Math. Scand. 44 (1979), 357–384.CrossRefGoogle Scholar
[4]Beauzamy, B. and Lapresté, J. T.. Modèles étalés des espaces de Banach. (Hermann, 1984).Google Scholar
[5]Bombal, F. and Fierro, C.. Compacidad débil en espacios de Orlicz de funciones vectoriales. Rev. Real Acad. Cienc. Exac. Fís. Natur. Madrid 78 (1984), 157–163.Google Scholar
[6]Bourgain, J.. An averaging result for l 1-sequences and applications to weakly conditionally compact sets in . Israel J. Math. 32 (1979), 289–298.CrossRefGoogle Scholar
[7]Bourgain, J.. The Komlós theorem for vector valued functions. Unpublished (1979).Google Scholar
[8]Diestel, J.. Sequences and series in Banach spaces (Springer, 1984).CrossRefGoogle Scholar
[9]Diestel, J. and Uhl, J. J. Jr. Vector measures Math. Surveys vol. 15 (Amer. Math. Soc., 1977).CrossRefGoogle Scholar
[10]Garling, D. J. H.. Subsequence principles for vector-valued random variables. Math. Proc. Cambridge Phil. Soc. 86 (1979), 301–311.CrossRefGoogle Scholar
[11]Guerre, S.. La propriété de Banach–Saks ne passe pas de E à L 2(E), d'après J. Bourgain. Séminaire d'Analyse Fonctionnelle 1979/80, Exposé no. 8, Ecole Polytechnique, Palaiseau.Google Scholar
[12]Kadec, M. I. and Pelczynsky, A.. Bases, lacunary sequences and complemented subspaces in the spaces Lp. Studia Math. 21 (1962), 161–176.CrossRefGoogle Scholar
[13]Komlós, J.. A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229.CrossRefGoogle Scholar
[14]Louveau, A.. Une méthode topologique pour l'etude de la propriété de Ramsey. Israel J. Math. 23 (1976), 97–116.CrossRefGoogle Scholar
[15]Partington, J. R.. On the Banach–Saks property. Math. Proc. Cambridge Phil. Soc. 82 (1977), 369–374.CrossRefGoogle Scholar
[16]Partington, J. R.. Almost sure summability of subsequences in Banach spaces. Studia Math. 71 (1981), 27–35.CrossRefGoogle Scholar
[17]Pisier, G.. Sur les espaces qui ne contiennent pas de uniformement. Séminaire Maurey-Schwartz 1973/74, Expose no. 7, Ecole Polytechnique, Paris.Google Scholar
[18]Schachermayer, W.. The Banach–Saks property is not L 2-hereditary. Israel J. Math. 40 (1981), 340–344.CrossRefGoogle Scholar
[19]Szlenk, W.. Sur les suites faiblement convergents dans l'espace L. Studia Math. 25 (1965), 337–341.CrossRefGoogle Scholar