Exercise on equivalence of weak convergence in Sobolev and $L^p$ spaces

Question

I have to solve the Exercise $10.43$ of page $302$ of the book "A First Course in Sobolev Space" by Leoni. The problem is the following.

Let $\Omega\subset\mathbb{R}^n$ be an open set, $1\leq p<+\infty$ and $\{u_n\}\subset W^{1, p}(\Omega)$. Prove that $u_n\rightharpoonup u$ in $W^{1, p}(\Omega)$ if and only if $u_n\rightharpoonup u$ in $L^p(\Omega)$ and $\nabla u_n\rightharpoonup\nabla u$ in $L^p(\Omega; \mathbb{R}^n)$.

Some ideas/hints?

Thank You

Answer 1

Note that $W^{1,p}(\Omega)\to L^p(\Omega)\times L^p(\Omega;\mathbb R^n):u\mapsto(u,\nabla u)$ is an isometry (perhaps after changing to an equivalent norm). Can you see what to do now?