Let $\{f_n\}$ be a sequence converging weakly in $C( [0,T] ; L^2(\mathbb{R}^n))$ and also converges in weak star sense in $L^\infty([0,T]; L^2(\mathbb{R}^n))$ to $f$. Can it be concluded that $f_n(t)$ converges weakly to $f(t)$ in $L^2(\mathbb{R}^n)$? And is the result true if instead, we have $C( [0,T] ; H^{-2}(\mathbb{R}^n))$?
I know in general there are counterexamples to this, but with the additional continuity in time condition, I think it should be possible.
Since the evaluation functional $Ef := f(t)$ is linear and continuous from $C([0,T],L^2)$ to $L^2$, it follows $Ef_n\rightharpoonup Ef$, which is the weak convergence $f_n(t)\rightharpoonup f(t)$ in $L^2$.