# Exercise 4.33 in Brezis functional analysis.

Fix a function $$\phi \in C_c(\mathbb{R}), \phi\not\equiv0,$$ and consider the family of functions

$$\mathcal{F} = \{\phi_n:n\in\mathbb N\},$$ where $$\phi_n(x) = \phi(x+n), x\in\mathbb{R}.$$

The problem is to prove that $$\mathcal{F}$$ does not have compact closure in $$L^p(\mathbb{R}) (1\le p <\infty),$$ but there is a theorem that the closure $$\mathcal{F}|_{\Omega}$$ in $$L^p(\Omega)$$ is compact for any finite-measure subset $$\Omega\in\mathbb{R}.$$ Thus this problem is intended to show that the theorem is not applicable to infinite-measure sets, in particular, $$\mathbb{R}.$$

I think I need to induce contradiction assuming $$\mathcal{F}$$ has compact closure in $$L^p(\mathbb{R}),$$ but I cannot come up with the next step.

Any hint or idea would be appreciated. Thanks in advance.

## 1 Answer

Hint: We have $$\operatorname{supp}\phi\subset [-N,N]$$ for some $$N\in\mathbb N$$. Can the sequence $$\{\phi_{nN}\}_{n\in\mathbb N}$$ have a convergent subsequence?

• Your answer makes me feel I am stupid... Thank you Aweygan! – Euduardo Nov 25 '18 at 5:33
• You're welcome, but don't feel stupid! – Aweygan Nov 25 '18 at 5:36