I wonder whether this fact is true or not (if a counter-example exists, please just give a hint on how to construct it!):


Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider a bounded sequence $\{f\_n\}$ in $L^\infty\left([0,T];L^2(\Omega)\right)$. Suppose I also know that $\{f\_n\}$ is in $L^2\left([0,T];L^2(\Omega)\right)$ and it also bounded there. Can I find some subsequence of 
$\{f\_n\}$ that converges in $L^\infty\left([0,T];L^2(\Omega)\right)$?




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Ok so I realized the question had an easy answer and thank you for the comments and answer! I decided to focus on the more important question which is **Generally, how does one study convergence in $L^\infty\left([0,T];L^2(\Omega)\right)$?**


I wish the answers would elaborate more on the weak$^\*$-convergence suggested in the comments. Also inspired by the comments, what tools from the Banach valued Bochner spaces theory can one use to help answering questions of convergence?



Convergence in $L^\infty([0,T];L^2(\Omega))$
It sounds like you are new to this subject, so there are some basic things for you to know. The first is that by night all Hilbert spaces are the same, so your $L^2(\Omega)$ can just be a generic $H$. Secondly, $L^\infty[0,T]$ is contained in $L^2[0,T]$ and the norm is smaller in $L^2$ up to a factor of $\sqrt{T}$, and nothing changes when you take values in $H$. So your sequence is automatically bounded in $L^2$, you don't need to assume that separately.


A third comment is that the standard sequence for counterexamples in $L^\infty[0,1]$ is $f\_n =$ the function which is constantly $0$ on the intervals $[\frac{2k}{2^n}, \frac{2k+1}{2^n})$ and constantly 1 on the complementary intervals. There's clearly no subsequence that converges in sup norm, though the sequence does converge weak\* to the function which is constantly $\frac{1}{2}$. As sharpend says in the comments, you will always be able to find a subsequence which converges in the weak\* topology (but you have to pay attention to how you define "weak\* topology" for Hilbert space-valued functions).


