I am trying to comprehend the proof of a theorem in the calculus of variations (sketch: the functional $\int\limits_\Omega f(Du_j(x))~dx$ is weak*-sequentially semicontinuous if and only if $f$ is quasiconvex), but I am stuck on a (probably) simple argument which is not explained there.
Let $f:\mathbb{R}^{m} \rightarrow \mathbb{R}$ be continuous and let $u_j \rightarrow u \text{ weak}^*$ in $L^\infty(\Omega,\mathbb{R}^m)$, $\Omega$ a bounded Lipschitz-domain. Does the following hold: $f(u_j) \rightarrow f(u)$ $\text{weak}^*$ $L^\infty(\Omega)$?
Here I identfied $L^{\infty}(\Omega,\mathbb{R}^m)$ with the dual of $L^{1}(\Omega,\mathbb{R}^m)$.
Edit: I added "$\text{weak}^*$" in the last line.