On the density of Banach spaces C(K) with the Grothendieck property

  title={On the density of Banach spaces C(K) with the Grothendieck property},
  author={Christina Brech},
  journal={arXiv: Functional Analysis},
  • C. Brech
  • Published 18 May 2006
  • Mathematics
  • arXiv: Functional Analysis
Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that "no nontrivial complemented subspace of a Grothendieck space C(K) is separable" cannot be strengthened by replacing "is separable" by "has density less than that of l ∞ ", without using an additional set-theoretic assumption. Such a… 
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