Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative

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

Let $A : Y \to Z$ and $B : X \to Y$ be continuous linear transformations.

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

Let $\circ$ denote the composition.

Then $\norm {\, \cdot \,}$ is submultiplicative:


 * $\norm {A \circ B} \le \norm A \cdot \norm B$