In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space. Then they write the inner product of [itex]|x\rangle[/itex] and [itex]|y\rangle[/itex] (say) by calling on the dual to [itex]|x\rangle[/itex]: [itex]\langle x | y\rangle[/itex]. There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space. How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.