Question regarding convergence is Sobolev spaces

Question

I have been reading Evan's book for PDE, and i have a question about the following definition

Definition. (i) Let $\{u_m\}_{m=1}^\infty, u \in W^{k,p}(U)$. We say $u_m$ converges to u in $W^{k,p}(U)$, written \begin{equation} u_m \to u \quad \text{in} \quad W^{k,p}(U), \end{equation} provided \begin{equation} \lim_{m \to \infty} ||u_m - u||_W^{k,p}(U) = 0. \end{equation} (ii) We write \begin{equation} u_m \to u \quad \text{in} \quad W_{loc}^{k,p}(U), \end{equation} to mean \begin{equation} u_m \to u \quad \text{in} \quad W^{k,p}(V), \end{equation} for each $V \subset \subset U$.

So what is the defintion trying to say 1. If we have convergence in $W_{loc}^{k,p}(U)$ will we have convergence on $W^{k,p}(U)$, or can we have a subset that is not necessary compact where convergence cannot happen.?

2. If we have (i) we will have (ii)?

So, can u help me with those two questions, and if u want to give me some extra info about something that u guys think that i should know concerning this definition please do it.

Thank yoy in advance

Answer 1

If $V \subseteq U$ then $\| u|_{V} \|_{W^{k,p}(V)} \leq \| u \|_{W^{k,p}(U)}$ so if $u_n$ converges to $u$ in $W^{k,p}(U)$ then $u_n|_{V}$ will converge to $u|_{V}$ in $W^{k,p}(V)$. Hence, converges in $W^{k,p}(U)$ implies converges in $W^{k,p}_{\text{loc}}(U)$ but it does not work the other way because a sequence might converge on each open subset $V$ which is compactly contained in $U$ but might fail to converge on the whole of $U$.

Answer 2

(i) implies (ii) since if $V\subset U$ then $\int_V|D^k u_m-Du|^p dx\le \int_U|D^k u_m-Du|^p dx\to 0$. On the other hand (ii) does not imply (i). Take $u_m=\frac1m$. Then on every open bounded set $V$ you will have $||u_m-0||_{W^{k,p}(V)}=\frac1m(\mathcal{L^1}(V))^{1/p}\to 0$ but $||u_m-0||_{W^{k,p}(\mathbb{R})}=\infty.$