I want to understand an implication made by M. Sofonea and A. Matei in their book "Variational Inequalities with Applications". The sequence $\lbrace u_n \rbrace_{n\in \mathbb{N}}$ be bounded in $L^\infty(0,T;X)$ and $\lbrace u_n'\rbrace_{n\in\mathbb{N}}$ be bounded in $L^2(0,T;X)$. Then we can extract a subsequence with
\begin{align*} u_n &\rightharpoonup^* u \text{ in } L^\infty(0,T;X) \\ u_n' &\rightharpoonup u' \text{ in } L^2(0,T;X). \end{align*}
Well, then the say, that
\begin{align*} u_n(t) \rightharpoonup u(t) \end{align*}
for all $t\in[0,T]$. Can someone give me a quote or a proof, why this holds? Thanks, FFoDWindow