Let $V \subset H$ be a dense and compact embedding. Let $$\lVert u_n\rVert_{L^\infty(0,T;H)} + \lVert u_n \rVert_{L^2(0,T;V)} < C$$ where $C$ is independent of $n$. It follows that eg. $u_n \rightharpoonup u$ in $L^2(0,T;V)$ and $u_n \rightharpoonup^* u$ in $L^\infty(0,T;H)$ for some $u$.

Does this imply $u_n \to u$ in $L^2(0,T;H)$ strongly?

I have seen this claim on page 11 of http://www.mat.unimi.it/users/rocca/cgquad2.ps (see equation 3.41) but I find it hard to believe. The reference cited there is a book by Lions in which I cannot find anything.

I would like a reference if thiis is true. Thanks.