Exercise 27, § 1

Let $E$ and $F$ be locally convex spaces. One says that a mapping $u\in \mathscr{L}(E; F)$ is weakly compact if it is a compact mapping from E into F endowed with the topology $\sigma (F,F')$.

(a) The weakly compact mappings form a subspace $\mathscr{L}^{fc}(E; F)$ of $\mathscr{L}(E; F)$. Let $E_1, F_1$ be locally convex spaces, $f\in \mathscr{L}(E_1; E)$, $g\in \mathscr{L}(F; F_1)$ and $u\in \mathscr{L}(E; F)$; if $u$ is weakly compact, then the same is true of $g\circ u\circ f$.

(b) One has $\mathscr{L}^c(E; F)\subset \mathscr{L}^{fc}(E; F)$. If E is an infinite-dimensional reflexive Banach space, the identity mapping of E is weakly compact but is not compact.