I would like help to clarify a possible misunderstanding that I have concerning to the beginning of the section $4$ of this lecture notes. I know that $V$ and $H$ are Hilbert spaces, $V \hookrightarrow H$, $H$ is identified as a subspace of $V'$ and the operator $L + tJ: V \longrightarrow V'$ is invertible. I think this operator is invertible by the Riesz Representation theorem once that $V$ is a Hilbert space, therefore $V$ and $V'$ are isomorphic, i.e., $V \cong V'$. The problem is that if we take $V = H^1(\Omega)$ and $H = L^2(\Omega)$, it is well-known that $H^1(\Omega) \subsetneq L^2(\Omega)$, but $L^2(\Omega) \subset (H^1(\Omega))' \cong H^1(\Omega)$, which implies that $L^2(\Omega) \subset H^1(\Omega)$ less isomorphism and contradicts $H^1(\Omega) \subsetneq L^2(\Omega)$. I think that I am wrong in some part of my reasoning, but I can not identify where. I would like help to identify it.
Thanks in advance!