\begin{definition}[Definition:Projection (Hilbert Spaces)]
Let $H$ be a Hilbert space.
Let $P \in \map B H$ be an idempotent operator.
Then $P$ is said to be a '''projection''' {{iff}}:
:$\ker P = \paren {\Img P}^\perp$
where:
:$\ker P$ denotes the kernel of $P$
:$\Img P$ denotes the image of $P$
:$\perp$ denotes orthocomplementation.
\end{definition}