Norm of Hermitian Operator/Corollary

Corollary to Norm of Hermitian Operator
Let $\mathbb F \in \set {\R, \C}$.

Let $\HH$ be a Hilbert space over $\mathbb F$.

Let $A : \HH \to \HH$ be a bounded Hermitian operator.

Let $\innerprod \cdot \cdot_\HH$ denote an inner product on $\HH$.

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

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