If $E$ is a complex Hilbert-space, then $A\in\Hom(E)$ is normal iff for all $x\in E$: $\|A^*x\| = \|Ax\|$.

We have for all $x\in E$: \[ \|A^*x\|^2 - \|Ax\|^2 = \langle A^*x,A^*x\rangle - \langle Ax,Ax\rangle = \langle AA^*x,x\rangle - \langle A^*Ax,x\rangle = \langle (AA^*-A^*A)x,x\rangle. \] Now if $A$ is normal, i.e. $AA^* = A^*A$, then we obtain $\|A^*x\|^2 - \|Ax\|^2 = 0$ and thus $\|A^*x\| = \|Ax\|$ for all $x\in E$. On the other hand, if $\|A^*x\|^2 - \|Ax\|^2 = 0$ for all $x\in E$, then by Lemma, applied to $AA^*-A^*A$, we obtain $AA^*-A^*A=0$, so $A$ is normal.