Theorem 4

— Let $F$ and $G$ be two Banach spaces, $u$ a continuous linear mapping of $F$ into $G$. For every function $f \in \mathscr{L}_F^p$, the composite function $u \circ f$ belongs to $\mathscr{L}_G^p$.

Let $f \in \mathscr{L}_F^p$; for every $\varepsilon > 0$, there exists a function $g \in \mathscr{K}_F$ such that $N_p(f - g) \leq \varepsilon$; since $|u \circ f - u \circ g| \leq \|u\| \cdot |f - g|$, we have

and since $u \circ g$ is continuous with compact support, the theorem is proved.