Exercise 19, § 2

Let E be a Banach space, E' its strong dual, G' a strongly closed subspace of E' satisfying the first axiom of countability for the strong topology. Show that there exists a countable subset of E such that, if F is the closed vector subspace of E generated by this subset, then G' is isometric to a strongly closed subspace of the strong dual F' of F. (Suppose that the sequence $(x'_n)$ is strongly dense in G'; for every n, let $x_n \in E$ such that $\|x_n\| \leq 1$ and $\langle x_n, x'_n \rangle = \left(1 - \frac{1}{n}\right)\|x'_n\|$; show that the strongly closed subspace F of E generated by the $x_n$ is the required space.)