Suppose $u_n\rightharpoonup u$ weakly in $W^{m,p}$ then show that $u_n$ converges to $u$ strongly in $W^{m,p}$.
I tried: If $u_n\rightharpoonup u$ weakly in $W^{m,p}$ then $f(D^{\alpha}u_n)\rightarrow f(D^{\alpha}u)$ for all $|\alpha|\leq m$, where $f$ is in Dual of $L^p$. So $$|f(D^{\alpha}u_n)-f(D^{\alpha}u)|\leq \|f\|\|D^{\alpha}u_n-D^{\alpha}u\|_{L^p}$$ But here i want the reverse of this inequality. Don't know how to proceed further. Thanks!!