Exercise 35, § 1

Let $E$ be a Banach space. One says that $E$ possesses the metric approximation property if the identity mapping of $E$ is adherent to the set of elements of norm $\leqslant 1$ of $\mathscr{L}^f(E)$ in the space $\mathscr{L}(E)$ endowed with the topology of compact convergence.

Prove that the following properties are equivalent:

(i) The space E possesses the metric approximation property ;

(ii) The unit ball of $\mathscr{L}(E)$ is the adherence, for the topology of compact convergence, of the set of elements of norm $\leqslant 1$ of $\mathscr{L}^f(E)$;

(iii) For every Banach space F, the unit ball of $\mathscr{L}(E; F)$ is the adherence, for the topology of compact convergence, of the set of elements of norm $\leqslant 1$ of $\mathscr{L}^f(E; F)$;

(iv) For every Banach space F, the unit ball of $\mathscr{L}(F; E)$ is the adherence, for the topology of compact convergence, of the set of elements of norm $\leqslant 1$ of $\mathscr{L}^f(F; E)$.