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A double-dual characterization of separable Banach spaces containingl 1

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Abstract

It is proved that a separable Banach spaceB contains a subspace isomorphic tol 1 if (and only if) there exists an element inB**, the double-dual ofB, which is not a weak* limit of a sequence of elements inB. ConsequentlyB contains an isomorph ofl 1 if (and only if) the cardinality ofB** is greater than that of the continuum.

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References

  1. R. Baire,Sur les Fonctions des Variables Réelles, 1899, pp. 16, 30.

  2. S. Banach,Théorie des Operations Linéaires, Monografje Matematyczne, Warsaw, 1932.

    Google Scholar 

  3. W. J. Davis, T. Figiel, W. B. Johnson and A. Pexczyński,Factoring weakly compact operators, J. Functional Analysis,17 (1974), 311–327.

    Article  MATH  MathSciNet  Google Scholar 

  4. L. E. Dor,On sequences spanning a complex l 1 space,47 (1975), 515–516.

    MATH  MathSciNet  Google Scholar 

  5. F. Hausdorff,Set Theory, Chelsea Publ. Co., New York, 1962.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. J. L. Kelley and I. Namioka,Linear Topological Spaces, D. Van Nostrand Co., Princeton, New Jersey, 1963, p. 118.

    MATH  Google Scholar 

  8. R. D. McWilliams,A note on weak sequential convergence, Pacific J. Math.12 (1962), 333–335.

    MATH  MathSciNet  Google Scholar 

  9. R. D. McWilliams,Iterated w*-sequential closure of a Banach space in its second conjugate, Proc. Amer. Math. Soc.14 (1963), 191–196.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. D. McWilliams,On iterated w*-sequential closure of cones, Pacific J. Math.38 (1971), 697–715.

    MathSciNet  Google Scholar 

  11. H. P. Rosenthal,A characterization of Banach spaces containing l 1, Proc. Nat. Acad. Sci. U.S.A.,71 (1974), 2411–2413.

    Article  MATH  MathSciNet  Google Scholar 

  12. H. P. Rosenthal,Pointwise compact subsets of the first Baire class, to appear.

  13. G. Choquet,Remarques à propos de la démonstration de l’unicité de P. A. Meyer, Séminaire Brelot-Choquet-Deny (Théorie de Potential),6 (1962), No. 8, 13 pp.

  14. R. Phelps,Lectures on Choquet’s Theorem, D. van Nostrand Co., Princeton, New Jersey, 1966.

    MATH  Google Scholar 

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Authors and Affiliations

  1. The Ohio State University, Columbus, Ohio, USA

    E. Odell & H. P. Rosenthal

  2. The University of California, Berkeley, Calif, USA

    E. Odell & H. P. Rosenthal

Authors
  1. E. Odell
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  2. H. P. Rosenthal
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Additional information

The research of the second-named author was partially supported by NSF — GP — 30798X1

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Odell, E., Rosenthal, H.P. A double-dual characterization of separable Banach spaces containingl 1 . Israel J. Math. 20, 375–384 (1975). https://doi.org/10.1007/BF02760341

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  • DOI: https://doi.org/10.1007/BF02760341

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