Kernel of Normal Operator is Kernel of Adjoint

Theorem
Let $H$ be a Hilbert space.

Let $A \in \map B H$ be a normal operator.

Then:
 * $\ker A = \ker A^*$

where:


 * $\ker$ denotes kernel
 * $A^*$ denotes the adjoint of $A$.

Proof
Let $x \in H$ be arbitrary.

Then:

Hence, by definition of set equality:


 * $\ker A = \ker A^*$