Definition:Projection (Hilbert Spaces)

Definition
Let $H$ be a Hilbert space.

Let $P \in B \left({H}\right)$ be an idempotent operator.

Then $P$ is said to be a projection iff:


 * $\operatorname{ker} P = \left({\operatorname{ran} P}\right)^\perp$

where $\operatorname{ker}$ denotes kernel, $\operatorname{ran}$ denotes range, and $\perp$ denotes orthocomplementation.