I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space $V$ induces a Hermitian inner product.
More explicitly if we identify $<\psi|\in V $ with $|\psi>\in V^*$ we define an inner product on $V$ by $(|\psi>,|\phi>)\rightarrow <\psi|\phi>$. This is basically what is written in the notes, and seems to be the way of doing things in QM, but coming from a pure background prompted me to consider its rigour.
Thinking more rigorously I was wondering under what conditions this is true in general?
It feels like I should be doing something like a reverse engineered Riesz Representation Theorem, but I can't work out exactly how. Sorry if this is a stupid/obvious question - I may be a little rusty after a week or so away from maths!
Edit: After a bit more reading and thinking it seems that the more rigorous way to go is the following. Assume that $V$ a Hilbert space, w.r.t. to an inner product which we'll notate by $<\psi|\phi>$. By the Riesz Repn Theorem we know that we may consider $<\psi|$ to be an element of the dual space $V^*$. My question now becomes, why can we assume $V$ a Hilbert space? What's the motivation for that in QM?
Thanks in advance.