Skip to main content
SpringerLink
Log in

Rosenthal’s l 1 Theorem

  • Chapter
Sequences and Series in Banach Spaces

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 92))

  • 2480 Accesses

Abstract

The Eberlein-Šmulian theorem tells us that in order to be able to extract from each bounded sequence in X a weakly convergent subsequence it is both necessary and sufficient that X be reflexive. Suppose we ask less. Suppose we ask only that each bounded sequence in X have a weakly Cauchy subsequence. [Recall that a sequence (x n ) in a Banach space X is weakly Cauchy if for each x* ∈ X* the scalar sequence (x*x n ) is convergent.] When can one extract from each bounded sequence in X a weakly Cauchy subsequence?

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beauzamy, B. 1979. Banach-Saks properties and spreading models. Math. Scand., 44, 357 – 384.

    MathSciNet  MATH  Google Scholar 

  2. Bessaga, C. and Pelczynski, A. 1958. A generalization of results of R. C. James concerning absolute bases in Banach spaces. Studia Math., 17, 165 – 174.

    MathSciNet  MATH  Google Scholar 

  3. Bourgain, J., Fremlin, D., and Talagrand, M. 1978. Pointwise compact sets of Baire-measurable functions. Amer. J. Math., 100, 845 – 886.

    Article  MathSciNet  MATH  Google Scholar 

  4. Diestel, J. and Seifert, C. J. 1979. The Banach-Saks ideal, I. Operators acting on C(Ω). Commentationes Math. Tomus Specialis in Honorem Ladislai Orlicz, 109–118, 343 – 344.

    Google Scholar 

  5. Dor, L. 1975. On sequences spanning a complex l1 space. Proc. Amer. Math. Soc., 47, 515 – 516.

    MathSciNet  MATH  Google Scholar 

  6. Fakhoury, H. 1977. Sur les espaces de Banach ne contenant pas l1(N). Math. Scand., 41, 277 – 289.

    MathSciNet  Google Scholar 

  7. Farahat, J. 1974. Espaces de Banach contenant lu d’apres H. P. Rosenthal. Seminaire Maurey-Schwartz (1973–1974), Ecole Polytechnique.

    Google Scholar 

  8. Hagler, J. 1973. Some more Banach spaces which contain l1. Studia Math., 46, 35–42.

    Google Scholar 

  9. Haydon, R. G. 1976. Some more characterizations of Banach spaces containing l1. Math. Proc. Cambridge Phil. Soc., 80, 269 – 276.

    Article  MathSciNet  MATH  Google Scholar 

  10. Heinrich, S. 1980. Closed operator ideals and interpolation. J. Funct. Anal., 35, 397 – 411.

    Article  MathSciNet  MATH  Google Scholar 

  11. James, R. C. 1974. A separable somewhat reflexive Banach space with nonseparable dual. Bull. Amer. Math. Soc., 80, 738 – 743.

    Article  MathSciNet  MATH  Google Scholar 

  12. Janicka, L. 1979. Some measure-theoretic characterizations of Banach spaces not containing l1 Bull. Acad. Polon. Sci., 27, 561 – 565.

    MathSciNet  MATH  Google Scholar 

  13. Lindenstrauss, J. and Stegall, C. 1975. Examples of separable spaces which do not contain l1 and whose duals are non-separable. Studia Math., 54, 81 – 105.

    MathSciNet  MATH  Google Scholar 

  14. Lohman, R. H. 1976. A note on Banach spaces containing l1. Can. Math. Bull., 19, 365 – 367.

    Article  MathSciNet  MATH  Google Scholar 

  15. Musial, K. 1978. The weak Radon-Nikodym property. Studia Math., 64, 151 – 174.

    MathSciNet  Google Scholar 

  16. Odell, E. and Rosenthal, H. P. 1975. A double-dual characterization of separable Banach spaces containing l1. Israel J. Math., 20, 375 – 384.

    Article  MathSciNet  MATH  Google Scholar 

  17. Pelczynski, A. 1968. On Banach spaces containing L1(μ). Studia Math., 30, 231 – 246.

    MathSciNet  MATH  Google Scholar 

  18. Pelczynski, A. 1968. On C(S) sub spaces of separable Banach spaces. Studia Math., 31, 513 – 522.

    MathSciNet  MATH  Google Scholar 

  19. Riddle, L. H. and Uhl, J. J., Jr. 1982. Martingales and the fine line between Asplund spaces and spaces not containing a copy of l1. In Martingale Theory in Harmonic Analysis and Banach spaces. Lecture Notes in Mathematics, Vol. 939. New York: Springer-Verlag, pp. 145 – 156.

    Google Scholar 

  20. Rosenthal, H. P. 1970. On injective Banach spaces and the spaces L ∞(μ) for finite measures μ. Acta Math., 124, 205 – 248.

    Article  MathSciNet  MATH  Google Scholar 

  21. Rosenthal, H. P. 1972. On factors of C[0,1] with nonseparable dual. Israel J. Math., 13, 361–378; Correction: ibid., 21(1975), 93 – 94.

    Google Scholar 

  22. Rosenthal, H. P. 1974. A characterization of Banach spaces containing l1. Proc. Nat. Acad. Sci. (USA), 71, 2411 – 2413.

    MATH  Google Scholar 

  23. Rosenthal, H. P. 1977. Pointwise compact subsets of the first Baire class. Amer. J. Math., 99, 362 – 378.

    Article  MathSciNet  MATH  Google Scholar 

  24. Rosenthal, H. P. 1978. Some recent discoveries in the isomorphic theory of Banach spaces. Bull. Amer. Math. Soc., 84, 803 – 831.

    Article  MathSciNet  MATH  Google Scholar 

  25. Talagrand, M. 1984 Weak Cauchy sequences in Ll(E). Amer. J. Math, to appear.

    Google Scholar 

  26. Weis, L. 1976. On the suijective (injective) envelope of strictly (co-) singular operators. Studia Math., 54, 285 – 290.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Math Sciences, Kent State University, 44242, Kent, OH, USA

    Joseph Diestel

Authors
  1. Joseph Diestel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1984 Springer-Verlag New York, Inc.

About this chapter

Cite this chapter

Diestel, J. (1984). Rosenthal’s l 1 Theorem. In: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, vol 92. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5200-9_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-5200-9_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-9734-5

  • Online ISBN: 978-1-4612-5200-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation