Range and Kernel of Idempotent are Algebraically Complementary

Theorem
Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then $\operatorname{ker} A$ and $\operatorname{ran} A$ are algebraically complementary, i.e.:


 * $\operatorname{ker} A \cap \operatorname{ran} A = \left({0}\right)$, the zero subspace
 * $\operatorname{ker} A + \operatorname{ran} A = H$, where $+$ signifies setwise addition.