Eigenvalues of Normal Operator have Orthogonal Eigenspaces

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a normal operator.

Let $\lambda, \mu$ be distinct eigenvalues of $A$.

Then $\operatorname{ker} \left({A - \lambda}\right) \perp \operatorname{ker} \left({A - \mu}\right)$.

Here $\operatorname{ker}$ denotes kernel, and $\perp$ signifies orthogonality.