suppose $\Omega \subset R^n$ is an open set,prove that $W^{1,p}(\Omega)$ is a complete normed linear space

Question

I only know that $W^{1,p}(\Omega)$ is a normed linear space,I don't know how to deal with the "complete" proof,any suggestions will be appreciated.

As someone mentioned,here the defining of $W^{k,p}(\Omega)$ is that:

$W^{k,p}(\Omega)=\{u\in L^p(\Omega)|D^{\alpha}u \in L^p(\Omega),|\alpha|<= k \}$

where $p>=1$,and for each $u\in W^{1,p}(\Omega)$,let $||u||_{W^{k,p}}=(\sum_{|\alpha|<=k}||D^{\alpha}u||_{p}^{p})^{\frac{1}{p}}$.

And what's more,the norm in this question is $||\cdot||_{W^{k,p}}$

Answer 1

Fellows,I found the equivalent form of this question.You can go to the Verifying that the Sobolev space is a Banach Space