Operator Zero iff Inner Product Zero

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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




Also see


Sources