General Theory of Partial differential equation

Question

I know that $W^{k,p}(\Omega)$ is a banach space as it is proved in mostly books but I need how to prove it simple form How start the proof of $W^{1,1} (\Omega)$ is a Banach space where $\Omega \subset \mathbf{R}^n$?

Answer 1

I'm guessing the main difficulty is in showing completeness, and not in showing that $W^{1,1}$ is a normed space. So, take $(f_n)$ be a cauchy sequence in $W^{1,1}(\Omega)$. Then it is cauchy in $L^1(\Omega)$. Therefore $f_n\to f$ for some $f\in L_1(\Omega)$. But we also see that $\nabla f_n$ is cauchy in $L_1(\Omega)$ so that $\nabla f_n\to g$ for some $g\in L_1(\Omega)$. It remains to show that $f$ belongs to $W^{1,1}(\Omega)$ with $\nabla f=g$. Indeed, for any$\varphi$ that is smooth and compactly supported on $\Omega$ we have (denoting the derivative of $f_n$ by $\nabla f_n$)

$$ \int_{\Omega}\varphi g=lim_n\int_{\Omega}\varphi\nabla f_n=lim_n\int_{\Omega}f_n\nabla\varphi= \int_{\Omega}f\nabla\varphi $$

This shows that $f$ is weakly differentiable with $\nabla f=g$ (since weak derivatives are unique). From here, it should be easy to see that $f_n\to f$ in $W^{1,1}(\Omega)$.