Is there a relation between weak-* convergence in $L^\infty((0,T); L^2(\Omega))$ and strong convergence in $L^2(\Omega)$ uniformly in $t$?

Question

I have problems in understanding a proof. So my question is:

Let $u_n$ be a sequence that converges weak-* in $L^\infty((0,T); L^2(\Omega))$ to $u$. Can I imply that $u_n$ converges strongly in $L^2(\Omega)$ uniformly in $t \in \left[0,T\right]$? Or is there something similiar I can imply?

Here, $\Omega$ is an open bounded subset of $\mathbb{R}^n$ and $T > 0$.