Exercise 7, § 3

Let $E$ and $F$ be Banach spaces, $u$ an element of $\mathscr{L}(E; F)$. Prove that $u$ is a Fredholm mapping if and only if the following conditions are satisfied:

(i) There exists a continuous seminorm $p$ on E and $C\geqslant 0$ such that the canonical mapping of E into the separated completion of $(E, p)$ is compact, and

for all $x\in E$;

(ii) There exists a continuous seminorm $q$ on $F'$ and $C'\geqslant 0$ such that the canonical mapping of $F'$ into the separated completion of $(F', q)$ is compact, and

for all $y'\in F'$.