Hello I would like to check if the proof of the following propostion is correct
Propostion Show that for all $f \in (H^m(\Omega))'$ exist $f_\alpha \in L^2(\Omega)$, $|\alpha|\leq m$ $$f(v)= \sum_{|\alpha|\leq m} \int_{\Omega} f_\alpha \partial^\alpha v dx \quad \forall v \in H^m(\Omega).$$ Proof: By Riesz representation theorem exist $\tilde v \in H^m(\Omega)$ \begin{align}f(v) &= \langle\tilde v, v \rangle_{H^m(\Omega)} \\ &= \sum_{|\alpha|\leq m} \int_{\Omega} \partial^\alpha \tilde v \cdot \partial^\alpha v\ dx \quad \forall v \in H^m(\Omega), \end{align} where $\partial^\alpha \tilde v \in L^2(\Omega)$ so we can choose $f_\alpha:=\partial^\alpha \tilde v$ for all $|\alpha|\leq m$ and the proof is complete.
The proof seems correct to me. More general you can show an analogous result for the case $p \neq 2$ using that there exists a canonical isometric isomorphism \begin{align} j \colon \left(L^p(\Omega)^{|\alpha|}\right)^{'} \to L^{p'}(\Omega)^{|\alpha|} \end{align} and the fact that $W^{m,p}(\Omega)$ can be isometrically embedded into $L^p(\Omega)^{|\alpha|}$, where $|\alpha| \leq m$.