Background: I work on a problem that deals with Sobolev spaces $H^s(\mathbb{R})$ of $L^2$-type. In my case $s$ is integer and $s\geq 3$. In order to evaluate some limits I need to show precompactness (or reletively compactness). The most simple definitions of precompact sets I know are:
Definition 1: A subset F of a space X is precompact in X if the closure of F is compact.
Definition 2: A subset F of a space X is precompact in X if every sequence in F has a subsequence that converge in X.
My question is: In the definitions above, if $F\equiv H^s(\mathbb{R})$, what could be the space X, i.e. what would be the example of a space where the $H^s(\mathbb{R})$ space is precompact in?
Instead of $\mathbb{R}$ it could be any of its subsets. And examples of compactness and weakly precompactness would be very useful to know also (if anyone knows them).
What I have found so far: in the book Robert Adams, John Fournier, Sobolev spaces, 2003, I found only precompactness that deals with the $L^p$ spaces but not $H^s$. Only exception was on the page 179 where it stands:"Any bounded set S in $W^{m,p}_0 (\Omega)$ is precompact in $L^p(\Omega)$". Also I know that many Sobolev spaces are reflexive, and by the Alaoglu's theorem any bounded subsets are weakly precompact.
Just to note that the possibilities for the space X that would go well with the problem I am working on, would be some subsets of the spaces $BV$, $\mathcal{M}$, $L^p$ for $p<\infty$, $\mathcal{D}^{,}$.
If you have any ideas or know any good reference in the literature about this, share it because I need help with this. Thanks in advance.