Exercise 2, § 1

Let $E$ and $F$ be Banach spaces over $K$. One says that $u\in \mathscr{L}(E; F)$ is completely continuous if it is adherent in $\mathscr{L}(E; F)$ to the space $\mathscr{L}^f(E; F)$ of linear mappings of finite rank. We denote by $\mathscr{L}^c(E; F)$ the space of completely continuous linear mappings from E into F, and put $\mathscr{L}^c(E) =\mathscr{L}^c(E; E)$. a) The space $\mathscr{L}^c(E)$ is a closed two-sided ideal of $\mathscr{L}(E)$. Let $u\in \mathscr{L}(E; F)$. The mapping $\ell \mapsto \ell \circ u$ is a continuous linear mapping from $F'$ into $E'$, denoted by $^tu$. If $u\in \mathscr{L}^c(E; F)$, then $^tu\in \mathscr{L}^c(F'; E')$.

b) Let J be a set and $F =\mathscr{C}_0(J)$. Let $u\in \mathscr{L}(E; F)$. For every $j\in J$, the mapping $\ell_j:x\mapsto u(x)(j)$ is an element of $E'$. We have $\|u\|=$ sup$_{j\in J}\|\ell_j\|$ and the mapping $u\mapsto (\ell_j)_{j\in J}$ induces by passing to the subspaces an isomorphism of $\mathscr{L}^c(E; F)$ into the Banach space $\mathscr{C}_0(J; E')$.

c) If K is locally compact, then $u\in \mathscr{L}(E; F)$ is completely continuous if and only if the image by $u$ of every bounded subset of E is relatively compact in F. (One may compare this result with the prop. 11 of III, p. 16.)