Composition of Continuous Linear Transformations is Continuous Linear Transformation

Theorem
Let $K$ be a field.

Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$, $\struct{Z, \norm {\, \cdot \,}_Z}$ be normed vector spaces over $K$.

$\map {CL} {X, Y}$ be the continuous linear transformation space.

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

For all $T \in \map {CL} {X, Y}$ and for all $S \in \map {CL} {Y, Z}$ let $ST : X \to Z$ be the composition of continuous linear transformations.

Then $ST \in \map {CL} {X, Z}$.

Continuity
We have that: