# $\ell_\infty$ is a Grothendieck space

The problem I am considering stated formally is this:

Show that if a sequence in $\ell_\infty^*$ is weak*-convergent, then it is also weakly convergent. We may reduce this to the case where the sequence is weak$^*$-null, and show that it is weakly null.

This is a special case of a result of Grothendieck from the 50's. It is an internal characterization of what is now called a Grothendieck space.

A Banach space $X$ is Grothendieck if every weak$^*$-convergent sequence in $X^*$ is weakly convergent.

In his "resume," Grothendieck proves that $C(K)$ for $K$ an extremally disconnected (also called Stonian) compact space satisfies this property. Since we can represent $\ell_\infty$ as $C(\beta\mathbb{N})$, the space of continuous functions on the Stone-Cech compactification of the natural numbers (which is Stonian), it satisfies this property.

So my questions are

1) If you know the general technique to show this property for $C(K)$ spaces as mentioned above could you give me a brief indication, and also

2) Is there something in this particular example that might make it easier to deal with than general $C(K)$?

Either way I assume we have to be able to either work blindly with elements of $\ell_\infty^{**}$ since we can't well characterize it, or there may be some other way to show weak convergence. (Or I also think we can say in more modern language that $\ell_\infty^{**}$ whatever it is is a von Neumann algebra if that is helpful at all).

• One reference: A proof of the statement is given on page 103 of Joseph Diestel's Sequences and Series in Banach Spaces. – David Mitra Apr 5 '12 at 15:30