Let consider $H^{-1}(\Omega)$ as the dual space of $H_0^1(\Omega)$, the classical Hilbert sobolev space with zero traces at $\partial \Omega$. Also, consider $\langle u,v \rangle_{H^{-1}(\Omega),H_0^1(\Omega)}$ as the duality between $H^{-1}(\Omega)$ and $H_0^1(\Omega)$.
Is this true that
$$ \langle u,v \rangle_{H^{-1}(\Omega),H_0^1(\Omega)} = \int_{\Omega} uv \, dx, \qquad \mathrm{for} \,\,\, u \in H^{-1}(\Omega) \cap L^{\infty}(\Omega), \,\,\, v \in H_0^1(\Omega). $$