On the action of $ (H^1(\Omega))^\prime$ on $H^1(\Omega) $, if the element of $ (H^1(\Omega))^\prime$ is in $L^2 (\Omega)$.

Question

Let $\Omega$ be any open set in $\mathbb{R}^n$. Let $(H^1(\Omega))^\prime$ denote the dual of $H^1(\Omega)$. Then do we have the following? \begin{equation*} H^1(\Omega) \subset L^2 (\Omega) \subset (H^1(\Omega))^\prime. \end{equation*} More specifically, can we say if $f \in L^2 (\Omega)$, then \begin{equation*} <f,v>_{(H^1(\Omega))^\prime, H^1(\Omega)} = \int_\Omega fv \qquad \forall v \in H^1 (\Omega). \end{equation*} I know the above identification holds for the triple $H_0^1(\Omega) \subset L^2 (\Omega) \subset H^{-1}(\Omega)$. But I am unsure about the case for $H^1(\Omega), L^2 (\Omega), (H^1(\Omega))^\prime$.