13
$\begingroup$

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces.

(See, e.g., this note for the proof of the Grothendieck property of these even duals, which uses the fact that these spaces are $C(K)$ spaces for large Stonian $K$-s.)

Is it known, in general, whether the second dual of a Grothendieck space is a Grothendieck space too? Is there a counterexample?

Edit (T. Kania). Let me record here for the future reference that this very question was asked by J. Diestel on p. 105 in

J. Diestel, "Grothendieck spaces and vector measures", Vector and Operator Valued Measures and Applications, Acad. Press (1973) pp. 97–108.

Improve this question
$\endgroup$
4
  • $\begingroup$ What is a Grothendieck space?? $\endgroup$
    – abx
    Nov 19, 2013 at 12:32
  • $\begingroup$ I'd just link here encyclopediaofmath.org/index.php/Grothendieck_space Grothendieck spaces are also a space ideal in the sense of Pietsch. $\endgroup$
    – Aleksei Lissitsin
    Nov 19, 2013 at 12:38
  • $\begingroup$ the same question on MSE $\endgroup$
    – Norbert
    Nov 23, 2013 at 11:44
  • $\begingroup$ Moved from edit: As Trond Abrahamsen pointed out, my alternative attempt "the claim can also be seen from the fact that they are M-ideals in their second duals, hence have Pełczyński's property (V) and therefore are Grothendieck spaces." at proving the Grothendieck property of even duals of $\ell_\infty$ fails at the first step, because non-reflexive dual spaces cannot be M-ideals in their second duals. $\endgroup$
    – Aleksei Lissitsin
    Jan 20, 2015 at 10:18

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.