Continuous Linear Transformation Algebra with Supremum Operator Norm is Normed Algebra

Theorem
Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.

Let $\map {CL} X := \map {CL} {X, X}$ be a continuous linear transformation space.

Let $\struct {\map {CL} X, *}$ be an associative algebra.

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

Then $\struct {\struct {\map {CL} X, *}, \norm {\, \cdot \,}}$ is a normed algebra.