I encountered the following, seemingly simple question: let $H^1(\mathbb{R})=W^{1,2}(\mathbb{R})$ be one of the Sobolev spaces. Suppose we have a sequence $(f_n)_{n\in\mathbb{N}}$ of functions in $H^1(\mathbb{R})$ which converge to some $f\in H^1(\mathbb{R})$ with respect to the standard $H^1(\mathbb{R})$-norm. Does this imply that $(f_n)_{n\in\mathbb{N}}$ also converges to $f$ in $L^\infty(\mathbb{R})$.
I already found the answer for this question for finite intervals and I also found that at least pointwise convergence holds. However, I really need uniform convergence on the unbounded interval. Thanks in advance!
Morrey's inequality states that $W^{1,p}(\mathbb{R}^n)$ is embedded into the Hölder space $C^{0,1-n/p}(\mathbb{R}^n)$.
In your case $n=1$ and $p=2$ so this shows that $H^1(\mathbb{R})$ is continuously embedded into $C^{0,1/2}(\mathbb{R})$, which is turn continuously embedded into $L^{\infty}(\mathbb{R})$ since the topology on the Hölder space is stronger than that of uniform convergence. In particular there exists a constant $C>0$ such that $$\|u\|_{\infty}\leq C\|u\|_{H^1}$$ for every $u\in H^1$, which shows that converging sequences w.r.t. to the $H^1$ norm also converge uniformly.
Note: This also works in higher dimensions and on more complicated domains than the whole of $\mathbb{R}^n$.