How to show that $\int_0^T (u_m(t), u_m(t))_{H} \to \int_0^T (u(t), u(t))?$ (Bochner space) Let $p \geq 1.$ Let $$W = \{ u \in L^p(0,T;V) : u' \in L^{q}(0,T;V^*)\}$$ 
where $V \subset H \subset V$ is a Gelfand triple ($V$ Banach and $H$ Hilbert), and $p$ and $q$ are conjugate indices. The space has the obvious norm $$|u|_W = |u|_{L^p(0,T;V)} + |u'|_{L^q(0,T;V^*)}.$$
Let $u \in W$, and let $u_m \in C^1([0,T];V)$ be a sequence with $$u_m \to u$$ in $W.$ How do I show that
$$\int_0^T (u_m(t), u_m(t))_{H} \to \int_0^T (u(t), u(t))$$?
I cannot seem to do this unless I restrict $p \geq 2$, which I want to avoid.

It is known that $W \hookrightarrow C([0,T]; H)$ and $C([0,T]; H) \hookrightarrow L^2(0,T; H)$ (with both embeddings being continuous). The (quadratic) functional
$$v \mapsto \int_0^T (v(t),v(t))_H \, dt$$
is just the squared norm in $L^2(0,T;H)$, hence it is continuous. Together with the above embeddings, this gives your desired convergence.
