\begin{definition}[Definition:Finite Rank Operator]
Let $H, K$ be Hilbert spaces.
Let $T: H \to K$ be a linear transformation.
Then $T$ is said to be a '''finite rank operator''', or '''of finite rank''', {{iff}} its range, $\Rng T$, is finite dimensional.
Note that a finite rank operator is not necessarily bounded.
\end{definition}