Let $H$ be a Hilbert space. Often people say "we identify $H$ with its dual $H^*$ with the Riesz representation theorem".
I know there is a map $R:H^* \to H$ which is ismometric and isomorphic. So by identifying, given an element $f \in H^*$, do we just replace $f$ with $Rf \in H$ in all formulaes and texts?
Really my question is, given a Hilbert space $H$, how do we know whether it has been identified with its dual or not?
A Hilbert space has been identified with its dual, if at places where one would expect an element from $H^*$ instead an element of $H$ is written. Let me explain this with some examples.
(1) Consider a function $f:H\to \mathbb R$. If $f$ is Gateaux differentiable, then its first derivative $f'(x)$ is an element in $H^*$: it maps a direction (in $H$) to an increment of $f$ (in $\mathbb R$), thus $f'(x) \in \mathcal L(H,\mathbb R)$. This is most prominent with $f(x) = \frac12\|x\|_H^2$. Then $f'(x)$ is given by $$ f'(x)v = \langle x,v\rangle. $$ If somebody writes the expression $f'(x) = x$ then you know that there is the identification of $H=H^*$ behind.
(2) In partial differential equations, you have the weak formulation $$ \int_\Omega \nabla y \nabla v dx = \int_\Omega f v \ dx. $$ This is often written as an equation in the dual of $H^1(\Omega)$: $$ -\Delta y = f. $$ Here, the function $f\in L^2(\Omega)$ has been identified with the integral functional $$ v\mapsto \int_\Omega f v \ dx, $$ which is an element of $H^1(\Omega)^*$ by construction. That is, here an identification of $f$ and the integral functional happened of with respect to the $L^2(\Omega)$ scalar product.
(3) Let $H_1,H_2$ be two Hilbert spaces, and $T\in \mathcal L(H_1,H_2)$ a linear and continuous operator. Then one can construct the adjoint operator $T^*\in\mathcal L(H_2^*,H_1^*)$. Now if somebody works with $T^*\in L(H_2,H_1)$ then (s)he uses the Hilbert-space adjoint $T^\star$ of $T$, where $T^\star = R_{H_1}T^*R_{H^2}^{-1}$. Here implicitly both Riesz isomorphisms of $H_1$ and $H_2$ are used. Note, that the notation of adjoint and Hilbert-space-adjoint may differ from textbook to textbook.
