# Definition:Projection (Hilbert Spaces)

## Definition

Let $H$ be a Hilbert space.

Let $P \in \map B H$ be an idempotent operator.

Then $P$ is said to be a projection if and only if:

$\ker P = \paren {\Img P}^\perp$

where:

$\ker P$ denotes the kernel of $P$
$\Img P$ denotes the image of $P$
$\perp$ denotes orthocomplementation.