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A non-reflexive Grothendieck space that does not containl ∞

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Abstract

A compact spaceS is constructed such that, in the dual Banach spaceC(S)*, every weak* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol ∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl ∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.

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References

  1. J. Diestel and J. J. Uhl,Vector Measures, American Mathematical Society, Providence, 1977.

    MATH  Google Scholar 

  2. S. Dineen,Bounding subsets of a Banach space, Math. Ann.192 (1971), 61–70.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. H. Fremlin,Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.

  4. J. Hagler,On the structure of S and C(S) for S dyadic, Trans. Amer. Math. Soc.214 (1975), 415–427.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Lindenstrauss and L. Tzafriri,Classical Banach, Spaces, Lecture notes in Mathematics338, Springer, Berlin-Heidelberg-New York, 1973.

    MATH  Google Scholar 

  6. H. P. Rosenthal,On relatively disjoint families of measures, with some applications, to Banach space theory, Studia Math.37 (1970), 13–36.

    MATH  MathSciNet  Google Scholar 

  7. W. Schachermeyer,On some classical measure-theoretical theorems for non-complete Boolean algebras, to appear.

  8. M. Schottenloher,Richness of the class of holomorphic functions on an infinite-dimensional space, inFunctional Analysis: Surveys and Recent Results (K.-D. Bierstedt and B. Fuchsteiner, eds.), North-Holland, 1977.

  9. G. L. Seever,Measures on F-spaces, Trans. Amer. Math. Soc.133 (1968), 267–280.

    Article  MATH  MathSciNet  Google Scholar 

  10. Z. Semadeni,Banach Spaces of Continuous Functions, PWN, Warszawa, 1971.

    MATH  Google Scholar 

  11. M. Talagrand,Un nouveau C(K) de Grothendieck, to appear.

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Authors and Affiliations

  1. Brasenose College, Oxford, England

    Richard Haydon

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  1. Richard Haydon
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Haydon, R. A non-reflexive Grothendieck space that does not containl ∞ . Israel J. Math. 40, 65–73 (1981). https://doi.org/10.1007/BF02761818

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  • DOI: https://doi.org/10.1007/BF02761818

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