Exercise 18, § 1

Let $E$, $F$, $G$ be Banach spaces and let $u$ be an element of $\mathscr{L}^c(E; F)$.

(a) Let $v$ be an element of $\mathscr{L}(E; G)$. Suppose that, for every $\varepsilon >0$, there exists $C>0$ such that

for every $x\in E$. Then $v\in \mathscr{L}^c(E; G)$.

(b) Let $w$ be an injective element of $\mathscr{L}(F; G)$. Then, for every $\varepsilon >0$, there exists $C>0$ such that

for every $x\in E$.