Range of Idempotent is Kernel of Complementary Idempotent

Theorem
Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then $\operatorname{ran} A = \operatorname{ker} \left({I - A}\right)$.

Corollary 1
Furthermore:


 * $\operatorname{ker} A = \operatorname{ran} \left({I - A}\right)$

Corollary 2
$\operatorname{ran} A$ is a closed linear subspace of $H$.