I'm trying to solve the following problem from Folland's book (exercise 9.36).
Suppose that $0 \neq \phi \in C_c^{\infty}(\mathbb{R}^n)$ and $\{a_j\}$ is a sequence in $\mathbb{R}^n$ with $|a_j|\rightarrow \infty$, and let $\phi_j(x)=\psi(x-a_j)$. Then $\{\phi_j\}$ is bounded in $H^s$ for every $s$ but has no convergent subsequence in $H^t$ for any $t$.
I could prove that the sequence is bounded and that has no convergent subsequence in $H^s$ for $s \geq 0$ by showing that given $m$, for $n \gg m$ (except for at most a finite quantity of $n$), \begin{equation} ||\phi_m-\phi_n||_{H^s} \geq ||\phi_m- \psi_n||_{L^2} \geq 2||\phi||_{L^2}. \end{equation}
For $s<0$ the first inequality in the equation above is not true and I can't find a way to solve it. Can any one help me?
I would recommend looking at weak convergence (perhaps it is technically more accurate to call what I am about to describe weak-star convergence, but since we are in a Hilbert space the two notions agree). Recall that the weak topology on $H^{-s}$ (here $s\geq0$, although the same definition works for $s<0$) is the topology on $H^{-s}$ arising from the family of seminorms $f\longmapsto\vert{\langle f,\cdot\rangle}\vert$ (where $\langle \cdot,\cdot\rangle$ denotes the inner product) where $f$ belongs to $H^{s}$.
Since $\{\varphi_n\}_n$ is bounded in $H^{-s}$ it suffices to check weak convergence on a dense subspace $V$ of $H^s$. Let $V= C^{\infty}_{c}$. Evidently, $V$ is dense. Furthermore, since for every compact subset $K\subset\mathbb{R}$ the sequence $\{\varphi_n|_K\}_n$ is eventually zero, it follows that $\lim_{n\to\infty}\langle{f,\varphi_n}\rangle=0$ for all $f\in V$. Thus, $\{\varphi_n\}_{n}$ converges to zero weakly. However, since the Sobolev norms of a function are unaffected by translation, it follows that $\{\Vert\varphi_n\Vert_{H^{-s}}\}_n$ is a constant sequence. Since $\varphi\neq0$ by assumption, it follows that no subsequence of $\{\varphi_n\}_n$ can converge in $H^{-s}$.
EDIT: As Jose has pointed out in the comments, the duality pairing I am using is not the inner product on $H^{-s}$. The proof still works though.