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[tex]\left(\langle\phi|A\right)|\psi\rangle=\langle\phi|\left(A|\psi\rangle\right)[/tex]

for all kets [itex]|\psi\rangle[/itex].

This is how the action of an operator on a bra vector was (roughly) described in Dirac's Principles, as well as in other texts that I've seen. Next, Dirac asserts that this "uniquely determines" [itex]\langle\phi|A[/itex].

I was trying to prove, or at least justify this claim, but to no avail. Nor have I seen a proof anywhere else. Can anyone help?