Operator Zero iff Inner Product Zero

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

Let $A \in \map B \HH$ be a bounded linear operator.

Suppose that:


 * $\forall h \in \HH: \innerprod {A h} h_\HH = 0$

Then $A$ is the zero operator.

Also see

 * Corollary to Norm of Hermitian Operator, a similar result.