Operator Zero iff Inner Product Zero

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

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

Suppose that:


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

Then $A$ is the zero operator.

Proof
For each $x, y \in \HH$ we have:

So:


 * $\innerprod {A x} y = 0$

for each $x, y \in \HH$.

Setting $y = A x$, we have:


 * $\norm {A x}^2 = 0$

by the definition of the inner product norm.

So from, we have $A x = 0$ for each $x \in \HH$.

Also see

 * Norm of Hermitian Operator/Corollary