I am reading Lawrence C. Evans's book "Partial Differential Equations". I am on page 259 in section 5.2, Sobolev spaces. He writes the following:
Let $\{u_m\}_{m\in\mathbb N},u\in W^{k,p}(U)$. We say $u_m$ converges to $u$ in $W^{k,p}(U)$, written $$u_m\to u ~~~~\text{in}~W^{k,p}(U)$$ Provided $$\lim_{m\to\infty} \Vert u_m-u\Vert_{W^{k,p}(U)}=0$$ We write $$u_m\to u~~~~\text{in}~W_{\text{loc}}^{k,p}(U)\tag{*}$$ to mean $$u_m\to u~~~~\text{in}~W^{k,p}(V)\tag{**}$$ For all $V\subset \subset U$.
NOTATION: $V\subset\subset U$ means that $V\subset \bar{V}\subset U$ and $\bar V$ is compact. $W^{k,p}(U)$ is a Sobolev space and $\Vert\cdot\Vert_{W^{k,p}(U)}$ is its associated norm.
My question:
Is there standard terminology for the type of convergence in $(*)$? To me the terminology "local convergence" seems fitting, but this appears not to be standard. It is quite close to compact convergence, however the convergence in each compact set $(**)$ in this case is not uniform - instead of convergence W.R.T the $L^\infty$ norm, it is convergence with respect to the $W^{k,p}$ norm. Is convergence in the $L^\infty$ and $W^{k,p}$ norms equivalent on compact sets? I know that pointwise and uniform convergence are not equivalent on compact sets.
Basically, is there standard terminology for "convergence in norm on compact subsets" ?