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Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I have a sequence $u_n$ satisfying

$$\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$$ for all $n$.

Can I get a weak-* convergent sequence?

I don't know if $L^\infty(0,T;L^\infty(\Omega))$ is the dual $L^1(0,T;L^1(\Omega))$.

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For a Banach space $B$, the space $L^1(0,T;B)$ is naturally identified with $L^\infty(0,T;B^*)$ if and only if $B^*$ has the Radon-Nikodym property. The space $L^\infty(\Omega)$ does not have the RNP.

In general, the dual of $L^1(0,T;B)$ is the larger space $\Lambda^\infty(0,T;B^*)$ which consists of weak*-measurable functions $f$ with $\|f\|_{B^*}\in L^\infty$. The weak* measurability means that for every $v\in B$, the scalar function $\langle v,f\rangle$ is measurable.

So, you can get a weak* convergent net to an element of $\Lambda^\infty(0,T;B^*)$. And a sequence if $L^1(0,T;B)$ is separable, which it is in your case.

As a freely available source, I recommend the lecture notes Martingales in Banach Spaces by Pisier.

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