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

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Also see

• Norm of Hermitian Operator/Corollary

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.2.15$