Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$.
A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists $w_n \in C^\infty([0,T];V)$ such that $w_n \to w$ in $W$.
How do I interpret that $w_n' \to w'$ in $L^2(0,T;V^*)$? Since $w_n' \in C^\infty([0,T];V)$ but $w' \in L^2(0,T;V^*)$. I know that $C^\infty([0,T];V) \subset \in L^2(0,T;V^*)$, but, since we are in a Gelfand triple system, should it really be $$|w_n' - w'|_{L^2(0,T;V^*)}^2 = \int_0^T |w_n'(t)-w'(t)|^2_{V^*} = \int_0^T |i^*(R(w_n'(t))-w'(t)|^2_{V^*}$$ where $R:H \to H^*$ is the Riesz isomorphism and $i^*:H^* \to V^*$ is the adjoint to the inclusion map from $V$ to $H$. Isn't that how to interpret this? Because now the equality on the RHS makes sense.