Operator is Hermitian iff Inner Product is Real

Theorem
Let $H$ be a Hilbert space over $\C$.

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

Then $A$ is self-adjoint iff:


 * $\forall h \in H: \left\langle{Ah, h}\right\rangle_H \in \R$