Exercise 34, § 1

Let $E$ be a Banach space. Prove that the following properties are equivalent:

(i) The space $E'$ is an approximation space ;

(ii) For every Banach space F, the closure of $\mathscr{L}^f(E; F)$ in $\mathscr{L}(E; F)$ is $\mathscr{L}^c(E; F)$;

(iii) For every closed subspace F of $c_0$, the closure of $\mathscr{L}^f(E; F)$ in $\mathscr{L}(E; F)$ is $\mathscr{L}^c(E; F)$.