Norm of Hermitian Operator/Corollary

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

Let $A \in \map B H$ be a self-adjoint operator.

Suppose also that:


 * $\forall h \in H: \sequence {A h, h}_H = 0$

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