Completeness of Sobolev space constructed from seminorm

Question

Define $W^{p,r}(\mathbb{R}^d):=\{f\in L^p(\mathbb{R}^d) : D^\alpha f\in L^p(\mathbb{R}^d), \forall 0<|\alpha|\le r\}$ where $1\le p\le\infty$. Let the seminorm on $W^{p,r}(\mathbb{R}^d)$ be $|f|_{W^{p,r}}:=\sum_{0<|\alpha|\le r}\Vert D^\alpha f\Vert_{L^p}$. Let $\mathcal{W}^{p,r}(\mathbb{R}^d):=\{[f]_\sim:f\in W^{p,r}(\mathbb{R}^d)\}$ where $[f]_\sim:=\{g\in W^{p,r}(\mathbb{R}^d):|f-g|_{W^{p,r}}=0\}$. Define $\Vert [f]_\sim\Vert_{\mathcal{W}^{p,r}}:=\vert f\vert_{W^{p,r}}$ as the norm on $\mathcal{W}^{p,r}$. Is $\mathcal{W}^{p,r}(\mathbb{R}^d)$ a Banach space? Basically, is $\mathcal{W}^{p,r}(\mathbb{R}^d)$ complete w.r.t. $\Vert\cdot\Vert_{\mathcal{W}^{p,r}}$?

Answer 1

In fact, what you define is $$ W^{p,r}(\mathbb{R}^d):=\{f\in L^p(\mathbb{R}^d) : D^\alpha f\in L^p(\mathbb{R}^d), \; 1\leq|\alpha|\le r\} $$ with the seminorm $$ |f|_{W^{p,r}}:=\sum_{1\leq|\alpha|\le r}\Vert D^\alpha f\Vert_{L^p}, $$ while $\mathcal{W}^{p,r}(\mathbb{R}^d)$ is the quotient space $W^{p,r}(\mathbb{R}^d)/\mathbb{R}$ with the quotient norm that coincides with $|f|_{W^{p,r}}$. To establish completeness of $W^{p,r}(\mathbb{R}^d)/\mathbb{R}$, just note that $W^{p,r}(\mathbb{R}^d)$ endowed with the norm $$ \|f\|_{W^{p,r}}:=\|f\|_{L^p(B)}+|f|_{W^{p,r}},\quad B\overset{\rm def}{=}\{x\in\mathbb{R}^d\colon\;|x|<1\}, $$ becomes a Banach space. For $r=1$, this is exactly the replica of the Sobolev space $L_p^1(\mathbb{R}^d)\overset{\rm def}{=}W^{p,1}(\mathbb{R}^d)$ in the original Sobolev notations.

Remark. Of course, the order of superscripts $\{p,r\}$ is to be reversed!