Range and Kernel of Idempotent are Algebraically Complementary

Theorem
Let $H$ be a Hilbert space.

Let $A$ be an idempotent operator.

Then $\ker A$ and $\Rng A$ are algebraically complementary, that is:


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