Range of Idempotent is Kernel of Complementary Idempotent

Theorem
Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then $\Rng A = \map \ker {I - A}$.

Corollary 1
Furthermore:


 * $\ker A = \Rng {I - A}$

Corollary 2
$\Rng A$ is a closed linear subspace of $H$.