I am trying to understand the proof of characterization of $H^{-1}(U)$ as in Evan's Partial Differential equations. My questions is different to that of this link.
In the theorem regarding the same, it is stated that
Assume $f \in H^{-1}$ then there exist $f^0$, $f^1$,....$f^n$ in $L^2(U)$ such that
\begin{equation} <f,v> = \int_{U}f^0v + \sum\limits_{i=1}^nf^iv_{x_i}dx \hspace10mm (v \in H^1_0)\nonumber \end{equation}
Here $⟨⋅,⋅⟩$ denotes the dual pairing of $H^{−1}$ and $H^{1}_0$.
In the proof, he states that if $f \in H^{−1}(U)$, then by Riesz representation theorem, there exists a unique function $u \in H^{1}_0(U)$, for all $v \in H^{1}_0(U)$ such that
\begin{equation} <f,v> = \int_UDu \cdot Dv + uvdx \nonumber. \end{equation} And hence it establishes the statement (i).
Following things confuse me:
If we are not supposed to identify $H^{−1}(U)$ with $H^{1}_0(U)$, how can we use the riesz representation theorem?
Since we have $L^2(U) \subset H^{−1}(U)$ how does that fit with the theorem. That is how does the fact that in general the inner product of $H^1_0$ might not be valid for a function in $L^2(U)$, which is not considered while invoking the Riesz theorem for the characterisation of an $f$ in $H^{-1}(U)$ in the above proof, fit with Riesz representation theorem?
How can we show any $f$, as defined above, can be attained in the limit as \begin{equation} <f,v> = \int_U u_nvdx \hspace2mm \forall v \in H^1_0(U) \end{equation} for some sequence $u_n$ in $L^2(U)$, since we know $L^2(U)$ is dense in $H^{−1}(U)$.
Lastly, what would we lose out on if we identified $H^{1}_0(U)$ with it's dual and what part of the Riesz representation theorem leaves room for some flexibility regarding identifying or not identifying $H^{1}_0(U)$ with it's dual.
