In my lecture notes of a class I took I found the following result. Unfortunately, I can't find a reference for the result. Does anyone have a reference or a hint where I could find the result?
Lemma: Let $(V,H,V^{\prime})$ be a Gelfand triple. Given $(u_n)_{n\geq 1}$ bounded in $L^2([0,T]; V) \cap L^{\infty}([0,T];H)$ and equicontinuous as $V^{\prime}$-valued functions and such that $u_n(0) \to u(0)$ strongly in $H$, one can extract a subsequence converging strongly in $L^2([0,T];H)$.
Any help is appreciated!
Best, Luke