Rellich compactness fails on ${\mathbb{R}}^n$ by translations

Given: Countable Choice; $n\ge 1$, $1\le p<\mathrm{\infty }$, and $\psi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ nonzero; for $k\ge 0$ put $u_k(x):=\psi (x-k{e}_1)$. Write $K:=\mathrm{supp}\psi$. A concrete choice is the bump of A Euclidean bump for a compact set inside an open set applied to the compact set $\{0\}$ inside the open unit ball: it is smooth, supported in $B(0,1)$, equal to $1$ at the origin and hence nonzero.

Classical derivatives of smooth compactly supported functions are weak derivatives. (Classical derivatives agree with weak derivatives)

Translation is an $L^p$-isometry and commutes with classical differentiation. $\mathrm{\parallel }{\tau }_{h}g{\mathrm{\parallel }}_{L^p}=\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p}$ for every $g\in L^p$ and every $h$, since Lebesgue measure is translation invariant, and ${\mathrm{\partial }}_j({\tau }_{h}g)={\tau }_h({\mathrm{\partial }}_{j}g)$ for smooth $g$. (Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation, Translation of a function on ${\mathbb{R}}^n$, The space $L^p(\mu )$ as the quotient by null functions)

Supports of separated translates are disjoint. $\mathrm{supp}{u}_k=K+k{e}_1$; if $\mathrm{\mid }k-l\mathrm{\mid }>2R$ where $K\subseteq B(0,R)$, then $(K+k{e}_1)\cap (K+l{e}_1)=\mathrm{\varnothing }$. A compact set is bounded, so such an $R$ exists. (The support of a function on ${\mathbb{R}}^n$ and its compactly supported Riemann integral, A compact subset of a metric space is closed and bounded, Translation of a function on ${\mathbb{R}}^n$)

The Sobolev norm. $\mathrm{\parallel }g{\mathrm{\parallel }}_{W^{1,p}({\mathbb{R}}^n)}=(\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p}^p+{\sum }_{j=1}^n\mathrm{\parallel }{\mathrm{\partial }}_{j}g{\mathrm{\parallel }}_{L^p}^p{)}^{1\mathrm{/}p}$. (Integer-order Sobolev spaces and their norms)