Norm of Hermitian Operator/Corollary

Corollary to Norm of Self-Adjoint Operator
Let $H$ be a Hilbert space.

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

Suppose also that:


 * $\forall h \in H: \left\langle{Ah, h}\right\rangle_H = 0$

Then $A$ is the zero operator $\mathbf 0$.