Theorem 1

— Let $F$ and $G$ be two Banach spaces, $u$ a continuous linear mapping of $F$ into $G$. For every integrable function $f$ with values in $F$, $u \circ f$ is integrable and

We already know that $u \circ f$ is integrable (§ 3, No. 5, Th. 4); the relation (6), being valid for every $f \in \mathscr{H}_F$, extends to every integrable function $f$ by the principle of extension of identities: for, $f \mapsto u \circ f$ is continuous for the topology of convergence in mean, as follows from the inequality $N_1(u \circ f) \leq \|u\| \cdot N_1(f)$.