I put beforehand that there are some similar questions in this blog, but I nonetheless would like to pose my question as I did not find any explanatory answer.
Let us consider a vector space H, equiped with an inner product: ( , ): $H\times H\ \rightarrow\ \mathbb{C}$. This is a sesquilinear positive definite form defined on the space H itself.
Now let us consider the dual space of H and denote it with H'. As in any topological vector space we can define a duality, through a bilinear form: $< , >: H' \times H\ \rightarrow\ \mathbb{C}$ which to any fanctional v and any vector w associates the evaluation v(w).
Now a Hilbert space, by means of Riesz-Frechet theorem is antisomorphic to its dual, thus there could be some connection between the two objects. The most I could find though is that if that we have an orthonormal basis (i.e. the space is separable) it should hold $<v',w>=(v,w)$ where by v' I mean the dual vector of v.
In fact consider an orthonormal basis $\{e_k\}$ and a vector $v^k e_k$ then its dual vector is a lnear form whihch acts on any vector $w^je_j$ yielding $v^kw^j e_k'(e_j)= v^kw^j\delta_{k,j}=v^kw^k$.
This should be the case in the space $L^2(V)$: my doubt was in fact caused by the quantum mechanichs case, since in my professor notes dual pairing and inner product are somehow identified.
But: Is my reasoning correct? What about spaces which do not admit an orthonormal basis? Is ther more to this connection?
Thanks.