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.

Let $S \circ T : X \to Z$ be the composition of mappings such that:


 * $\forall T \in \map {CL} {X, Y} : \forall S \in \map {CL} {Y, Z} : \forall x \in X : \map {S \circ T} x := \map S {\map T x}$

Then $S \circ T \in \map {CL} {X, Z}$.

Linearity
Follows from Composition of Linear Transformations is Linear Transformation.

Continuity
We have that:

By Continuity of Linear Transformation between Normed Vector Spaces, $S \circ T$ is continuous.