# Weak star convergence

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