Space of Bounded Linear Transformations is Banach Space

Theorem
Let $H, K$ be Hilbert spaces.

Let $\map B {H, K}$ denote the space of bounded linear transformations from $H$ to $K$.

Let $\Bbb F \in \set {\R, \C}$ denote the ground field of $K$.

Now $\map B {H, K} \subseteq K^H$, the set of mappings from $H$ to $K$.

Therefore, $\map B {H, K}$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$).

Let $\norm{\,\cdot\,}$ denote the norm on bounded linear transformations.

Then $\norm{\,\cdot\,}$ is a norm on $\map B {H, K}$.

Furthermore, $B \left({H, K}\right)$ is a Banach space with respect to this norm.