property of local Sobolev space

Question

The local Sobolev space,defined as $W^{k,p}_{loc}(\Omega)$, is the space such that for any $u \in W^{k,p}_{loc}(\Omega)$ and any compact $V\subset \Omega$, $u \in W^{k,p}(V)$. I am just wondering if for any $u\in W^{k,p}_{loc}(\Omega)$, there exists ${u_n}\in C^\infty(\Omega)$, such that $u_n \rightarrow u$ in $W^{k,p}(V)$ for any compact $V\subset \Omega$

Answer 1

Yes. Take a sequence of smooth cutoff functions $\eta_n$, each with compact support in $\Omega$, and such that every compact subset of $\Omega$ is contained in the set $\{\eta_n=1\}$ for all sufficiently large $n$.

The function $u \eta_n$ belongs to $W^{k,p}(\Omega)$, and therefore there is a function $u_n\in C^\infty(\Omega)$ such that $\|u\eta_n-u_n\|_{W^{k,p}(\Omega)}<1/n$. The sequence $(u_n)$ does what you want.