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 :


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

where:
 * $\ker P$ denotes the kernel of $P$
 * $\operatorname{Im} P$ denotes the image of $P$
 * $\perp$ denotes orthocomplementation.

Also see

 * Orthogonal Projection is Projection
 * Characterization of Projections