Skip to main content
SpringerLink
Log in

On the subspaces ofL p(p>2) spanned by sequences of independent random variables

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL p, each of mean zero, is shown to be isomorphic tol 2,l p,l 2⊕l p, or a new spaceX p , and the linear topological properties ofX p are investigated. It is proved thatX p is isomorphic to a complemented subspace ofL p and another uncomplemented subspace ofL p, whence there exists an uncomplemented subspace ofl p isomorphic tol p. It is also proved thatX p is not isomorphic to the previously knownℒ p spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Bessaga and A. Pelczynski,On bases and unconditional convergence of series in Banach spaces, Studia Math.17 (1958), 151–164.

    MATH  MathSciNet  Google Scholar 

  2. M. M. Day,Normed Linear Spaces, Berlin, Springer-Verlag, 1958.

    MATH  Google Scholar 

  3. N. Dunford and J. T. Schwartz,Linear Operators I, New York, Intersci. Pub., 1958.

    MATH  Google Scholar 

  4. M. I. Kadec and A. Pelczynski,Bases, lacunary sequences, and complemented subspaces in the spaces L p , Studia Math.21 (1962), 161–176.

    MATH  MathSciNet  Google Scholar 

  5. J. Khinchine,Über die diadischen Brüche, Math. Z.18 (1923), 109–116.

    Article  MathSciNet  Google Scholar 

  6. J. Lindenstrauss,Extension of compact operators, Mem. Amer. Math. Soc.48 (1964).

  7. J. Lindenstrauss and A. Pelczynski,Absolutely summing operators in ℒ p spaces and their applications, Studia Math.29 (1968), 275–326.

    MATH  MathSciNet  Google Scholar 

  8. J. Lindenstrauss and A. Pelczynski,Contributions to the theory of the classical Banach spaces, to appear.

  9. J. Lindenstrauss and H. P. Rosenthal,Automorphisms in c 0,l 1,and m, Israel J. Math.7 (1969), 227–239.

    MATH  MathSciNet  Google Scholar 

  10. J. Lindenstrauss and H. P. Rosenthal,The ℒ p spaces, Israel J. Math.7 (1969), 325–349.

    MATH  MathSciNet  Google Scholar 

  11. M. Loeve,Probability Theory, Princeton, D. Van Nostrand, 1962.

    Google Scholar 

  12. R. E. A. C. Paley,A remarkable series of orthogonal functions, Proc. London Math. Soc.34 (1932), 241–264.

    Article  MATH  Google Scholar 

  13. A. Pelczynski,On the isomorphism of the spaces m and M, Bull. Acad. Polon. Sci., Sér. Math., Astronom. Phys.,6 (1958), 695–696.

    MATH  MathSciNet  Google Scholar 

  14. A. Pelczynski,Projections in certain Banach spaces, Studia Math.19 (1960), 209–228.

    MATH  MathSciNet  Google Scholar 

  15. A. Pelczynski and I. Singer,On non-equivalent bases and conditional bases in Banach spaces, Studia Math.25 (1964), 5–25.

    MATH  MathSciNet  Google Scholar 

  16. H. P. Rosenthal,On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from L p(μ)to L r (v), J. Functional Analysis2 (1969), 176–214.

    Article  Google Scholar 

  17. H. P. Rosenthal,On totally incomparable Banach spaces, J. Functional Analysis2 (1969), 167–175.

    Article  Google Scholar 

  18. H. P. Rosenthal,Projections onto translation-invariant subspaces of L p (G), Mem. Amer. Math. Soc.63 (1966).

  19. I. Singer,Some characterizations of symmetric bases in Banach spaces, Bull. Acad. Polon. Sci., Ser. Math. Astronom Phys.,10 (1962), 185–192.

    MATH  Google Scholar 

  20. A. Sobczyk,Projections in Minkowski and Banach spaces, Duke Math. J.8 (1944), 78–106.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, Berkeley, California

    Haskell P. Rosenthal

Authors
  1. Haskell P. Rosenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The work for this research was partially supported by the National Science Foundation GP-12997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rosenthal, H.P. On the subspaces ofL p(p>2) spanned by sequences of independent random variables. Israel J. Math. 8, 273–303 (1970). https://doi.org/10.1007/BF02771562

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02771562

Keywords

Search

Navigation