Proposition 8

Let $X, Y, Z$ be three normed spaces over $K$. The bilinear mapping $(u, v) \to v \circ u$ of $\mathscr{L}(X; Y) \times \mathscr{L}(Y; Z)$ into $\mathscr{L}(X; Z)$ is continuous.

For if $u \in \mathscr{L}(X; Y)$ and $v \in \mathscr{L}(Y; Z)$ we have

since for all $x \in X$ we have $||v(u(x))|| \leq ||v|| \cdot ||u(x)|| \leq ||v|| \cdot ||u|| \cdot ||x||$ by reason of (2).

In particular, on the set $\mathscr{L}(X)$ of continuous endomorphisms of a normed space $X$ over $K$, the norm $||u||$ is compatible with the $K$-algebra structure of $\mathscr{L}(X)$.