Finite Rank Operators Dense in Compact Linear Transformations

Theorem
Let $H, K$ be Hilbert spaces.

Then:
 * $B_{00} \left({H, K}\right)$ is everywhere dense in $B_0 \left({H, K}\right)$

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

That is, for every $T \in B_0 \left({H, K}\right)$, there is a sequence $\left({T_n}\right)_{n \in \N}$ in $B_{00} \left({H, K}\right)$ such that:
 * $\displaystyle \lim_{n \to \infty} \left\Vert{T_n - T}\right\Vert = 0$

where $\left\Vert{\cdot}\right\Vert$ denotes the operator norm.