Exercise 11, § 5

Let $A$ be an infinite set, and let $E = \overline{\mathscr{H}}(A)$ (IV, p. 47, exerc. 1), $E' = \ell^1(A)$ its dual, $E'' = \ell^\infty(A)$ its bidual (IV, p. 54, exerc. 14). Show that every subset of $E'$, which is compact for $\sigma(E', E'')$ is strongly compact (use Šmulian’s theorem and IV, p. 54, exerc. 15).