Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-20T06:58:22.819Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on symmetric basic sequences in Lp(Lq)

Published online by Cambridge University Press:  24 October 2008

Yves Raynaud
Yves Raynaud
Affiliation:
Equipe d'Analyse (CNRS), Université Paris-6, 4, place Jussieu, 75252-PARIS-Cedex 05, France
Get access Rights & Permissions [Opens in a new window]

Extract

Subspaces of Lp spanned by symmetric independent identically distributed random variables were identified as Orlicz spaces by Bretagnolle and Dacunha-Castelle[1], who showed that, conversely, in the case p ≤ 2, every p-convex, 2-concave Orlicz space is isomorphic to a subspace of Lp. This was extended by Dacunha-Castelle [3] to subspaces of Lp with symmetric basis, which appear as ‘p-means’ of Orlicz spaces (see [9] for the corresponding finite-dimensional result, and [12] for the case of rearrangement invariant function spaces). On the contrary the only subspaces with symmetric basis of Lp for p ≥ 2 are lp and l2 (if one does not care about isomorphy constants).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 1 , July 1992 , pp. 183 - 194
Copyright
Copyright © Cambridge Philosophical Society 1992

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]Bretagnolle, J. and Dacunha-Castelle, D.. Applications de l'étude de certaines formes linéaires aléatoires au plongement d'espaces de Banach dans des espaces Lp. Ann. Sci. École Norm. Sup. (4) 2 (1969), 437–480.CrossRefGoogle Scholar
[2]Bergh, J. and Löfström, J.. Interpolation Spaces, an Introduction. Grundlehren der math. Wiss. vol. 223 (Springer-Verlag, 1976).CrossRefGoogle Scholar
[3]Dacunha-Castelle, D.. Variables aléatoires échangeables et espaces d'Orlicz. In Séminaire Maurey-Schwartz 1974–75, exposés 10–11 (Ecole Polytechnique, 1975).Google Scholar
[4]Dacunha-Castelle, D.. Sous-espaces symétriques des espaces d'Orlicz. In Séminaire de Probabilités de Strasbourg IX (1973–74), Lecture Notes in Math. vol. 465 (Springer-Verlag, 1975), pp. 268–294.Google Scholar
[5]Garling, D. J. H.. Stable Banach spaces, random measures and Orlicz function spaces. In Probability Measures on Groups, Lecture Notes in Math. vol. 928 (Springer-Verlag, 1982), pp. 121–175.CrossRefGoogle Scholar
[6]Hoffmann-Jorgensen, J.. Sums of independent Banach space valued random variables. Studia Math. 52 (1974), 159–186.CrossRefGoogle Scholar
[7]Jain, N. C. and Marcus, M. B.. Integrability of infinite sums of independent vector-valued random variables. Trans. Amer. Math. Soc. 212 (1975), 1–36.CrossRefGoogle Scholar
[8]Krivine, J.-L. and Maurey, B.. Espaees de Banach stables. Israel J. Math. 39 (1981), 273–295.CrossRefGoogle Scholar
[9]Kwapień, S. and Schütt, C.. Some combinatorial and probabilistic inequalities and their application to Banach Space Theory. Studia Math. 82 (1985), 91–106.CrossRefGoogle Scholar
[10]Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, vol. 2. Ergebnisse der Math. vol. 97 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[11]Raynaud, Y.. Sur les sous-espaces de Lp(Lq). In Séminaire de Géométrie des Espaces de Banach Paris VI–VII, 1984–85, Publ. Math. Univ. Paris VII vol. 26, pp. 49–71.Google Scholar
[12]Raynaud, Y. and Schütt, C.. Some results on symmetric subspaces of L 1. Studia Math. 89 (1988), 27–35.CrossRefGoogle Scholar
[13]Rosenthal, H. P.. On the subspaces of Lp (p > 2) spanned by sequences of independent random variables. Israel J. Math. 8 (1970), 273–303.CrossRefGoogle Scholar