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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.

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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 $L^2$.

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  • $\begingroup$ How do we use the second info about $L^\infty$? $\endgroup$ – Goal123 May 1 at 17:55
  • $\begingroup$ Never. This weak-star convergence is weaker than the weak convergence in $C([0,T],L^2)$. $\endgroup$ – daw May 1 at 17:57
  • $\begingroup$ I think it is used to guess the weak limit of $f_n$ , since we are not given any candidate. $\endgroup$ – Goal123 May 1 at 18:10

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