Finite Rank Operators Dense in Compact Linear Transformations

Theorem
Let $H, K$ be Hilbert spaces.

Then:
 * $\map {B_{00} } {H, K}$ is everywhere dense in $\map {B_0} {H, K}$

where:
 * $\map {B_{00} } {H, K}$ is the space of continuous finite rank operators from $H$ to $K$
 * $\map {B_0} {H, K}$ is the space of compact linear transformations from $H$ to $K$.

That is, for every $T \in \map {B_0} {H, K}$, there is a sequence $\sequence {T_n}_{n \mathop \in \N}$ in $\map {B_{00} } {H, K}$ such that:
 * $\ds \lim_{n \mathop \to \infty} \norm {T_n - T} = 0$

where $\norm {\, \cdot \,}$ denotes the norm on bounded linear transformations.