Operator Zero iff Inner Product Zero

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

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

Suppose that:


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

Then $A$ is the zero operator.

Also see

 * Corollary to Norm of Self-Adjoint Operator, a very similar result.