In Quantum Mechanics, we have linear operators which can act on a ket to produce a new ket. However, we also allow the same operators to act on a bra vector to produce a new bra vector. That is, if [itex]\langle\phi|[/itex] is a bra and A is an operator, the action of A on [itex]\langle\phi|[/itex] is to produce a new bra denoted by [itex]\langle\phi|A[/itex]. Furthermore, we demand that [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?