A sequence $\{u_n\}$ in $W^{1,p}(1 <p \leq \infty)$ converges to some $u$ in $L^p$, and $\{u_n'\}$ is bounded in $L^p$. How to prove $u \in W^{1,p}$?

Question

It's a remark in Haim Brezis's Functional Analysis without proof. I have thought it for a long time, is there any hint?

Answer 1

You can find a subsequence such that $u_{n_k}'\rightharpoonup w$ in $L^p$ for $p<\infty$, convergence is $\rightharpoonup^*$ in $L^\infty$. Using the definition of weak derivative, one can prove $v=u'$.