Let $U \subset \mathbb{R}^n$ be a bounded domain.
Let $$W=\{ u \in L^2(0,T;H^1(U))\cap L^\infty(0,T;L^\infty(U)) : u' \in L^2(0,T;H^{-1}(U))\}$$ and let $$D=\{u \in L^2(0,T;H^1(U)) \cap L^\infty(0,T;L^\infty(U)) : u' \in L^2(0,T;L^2(U))\}.$$
Is it possible that $D$ is dense in $W$ with respect to the norm $$\lVert u \rVert = \lVert u \rVert_{L^2(0,T;H^1)} + \lVert u' \rVert_{L^2(0,T;H^{-1})}?$$
How about with the the $L^\infty(0,T;L^\infty)$ norm included above too?