Continuous Linear Transformation Space as Banach 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 $*_X : X \times X \to X$ and $* : \map {CL} X \times \map {CL} X \to \map {CL} X$ be bilinear mappings.

Suppose $\struct {\struct {X, \norm {\, \cdot \,}_X}, *_X}$ is a Banach algebra.

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

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