In Sakurai's Modern Quantum Mechanics, he develops the Dirac notation of bras and kets. In one part, he states (page 17): <B|X|A> = (<A|X^|B>)* = <A|X^|B>* where X^ denotes the Hermitian adjoint (the conjugate transpose) of the operator X. My question is, since a bra is the conjugate transpose of a ket, could we write <B|X|A>^ = <A|X|B> (since of course X^ = X (ie. X is Hermitian) for real, measurable quantities). What I'm trying to ask is, does Sakurai define a separate conjugate transpose of the bras and kets, whereby he just takes the complex conjugate and implicitly also transposes it (ie. |A>* = <A| ). Normally, I would expect to see |A>^ = <A|.