I am reading the paper "The Kolmogorov-Riesz Compactness Theorem", which gives a characterisation for totally bounded subsets of $L^P(\mathbb{R}^n)$ for $1 \le p <\infty$.
A subset $\mathcal{F} \subset L^p(\mathbb{R}^n)$ is totally bounded iff
i) $\mathcal{F}$ is bounded.
ii) $\forall \varepsilon>0, \ \exists\ R$ such that, for every $f \in \mathcal{F}$ $$\int_{|x|>R} |f(x)|^p\,dx < \varepsilon^p ,$$
iii) $\forall \varepsilon >0, \exists\ \rho >0$ such that for every $f \in \mathcal{F}$ and $y \in \mathbb{R}^n$ with $|y|< \rho$ $$\int_{\mathbb{R}^n} |f(x+y) -f(x)|^p\ dx <\varepsilon^p . $$
I was trying to find a subset of $L^p(\mathbb{R}^n)$ which is closed and bounded but not compact. For this I need a set which does not meet the criteria ii) or iii). Then it will not be totally bounded and hence not compact. Any help in finding this type of set?
The closed unit ball in a normed space is closed and bounded but compact if and only if the space is finite dimensional. You can find a proof that uses Riesz's lemma here.