Space of Compact Linear Transformations is Banach Space

Theorem
Let $H, K$ be Hilbert spaces.

Let $\map {B_0} {H, K}$ be the space of compact linear transformations from $H$ to $K$.

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

Now $\map {B_0} {H, K} \subseteq K^H$, the set of mappings from $H$ to $K$.

Therefore, $\map {B_0} {H, K}$ can be endowed with pointwise addition ($+$) and ($\Bbb F$)-scalar multiplication ($\circ$).

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

Then $\norm {\, \cdot \,}$ is a norm on $\map {B_0} {H, K}$.

Furthermore, $\map {B_0} {H, K}$ is a Banach space with respect to this norm.