Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space/Corollary 1

Theorem
Let $\mathbb K = \set {\R, \C}$.

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

Let $CL$ be the continuous linear transformation space.

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

Then $\struct {\map {CL} {X, \mathbb K}, \norm {\, \cdot \,} }$ is a Banach space.