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 $L^\infty\left([0,T];L^2(\Omega)\right)$?

Generally, how does one study convergence in $L^\infty\left([0,T];L^2(\Omega)\right)$?

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 $L^\infty\left([0,T];L^2(\Omega)\right)$?

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?