Israel Journal of Mathematics

, Volume 40, Issue 1, pp 65–73 | Cite as

A non-reflexive Grothendieck space that does not containl

  • Richard Haydon


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Diestel and J. J. Uhl,Vector Measures, American Mathematical Society, Providence, 1977.MATHGoogle Scholar
  2. 2.
    S. Dineen,Bounding subsets of a Banach space, Math. Ann.192 (1971), 61–70.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. H. Fremlin,Topological Riesz Spaces and Measure Theory, Cambridge University Press, 1974.Google Scholar
  4. 4.
    J. Hagler,On the structure of S and C(S) for S dyadic, Trans. Amer. Math. Soc.214 (1975), 415–427.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Lindenstrauss and L. Tzafriri,Classical Banach, Spaces, Lecture notes in Mathematics338, Springer, Berlin-Heidelberg-New York, 1973.MATHGoogle Scholar
  6. 6.
    H. P. Rosenthal,On relatively disjoint families of measures, with some applications, to Banach space theory, Studia Math.37 (1970), 13–36.MATHMathSciNetGoogle Scholar
  7. 7.
    W. Schachermeyer,On some classical measure-theoretical theorems for non-complete Boolean algebras, to appear.Google Scholar
  8. 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.Google Scholar
  9. 9.
    G. L. Seever,Measures on F-spaces, Trans. Amer. Math. Soc.133 (1968), 267–280.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Z. Semadeni,Banach Spaces of Continuous Functions, PWN, Warszawa, 1971.MATHGoogle Scholar
  11. 11.
    M. Talagrand,Un nouveau C(K) de Grothendieck, to appear.Google Scholar

Copyright information

© Hebrew University 1981

Authors and Affiliations

  • Richard Haydon
    • 1
  1. 1.Brasenose CollegeOxfordEngland

Personalised recommendations