Fréchet-Kolmogorov theorem

In functional analysis, the Fréchet-Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an L^p space. It can be thought of as an L^p version of the Arzelà-Ascoli theorem, from which it can be deduced.

Statement

Let $B$ be a bounded set in $L^p(\mathbb{R}^n)$ , with $p\in[1,\infty)$ .

The subset B is relatively compact if and only if the following properties hold:

$\lim_{r\to\infty}\int_{|x|>r}\left|f\right|^p=0$ uniformly on B,
$\lim_{a\to 0}\Vert\tau_a f-f\Vert_{L^p(\mathbb{R}^n)} = 0$ uniformly on B,

where $\tau_a f$ denotes the translation of $f$ by $a$ , that is, $\tau_a f(x)=f(x-a) .$

The second property can be stated as $\forall \varepsilon >0 \, \, \exists \delta >0$ such that $\Vert\tau_a f-f\Vert_{L^p(\mathbb{R}^n)} < \varepsilon \, \, \forall f \in B, \forall a$ with $|a|<\delta .$

References

Brezis, Haïm (2010). Functional analysis, Sobolev spaces, and partial differential equations. Universitext. Springer. p. 111. ISBN 978-0-387-70913-0.
Marcel Riesz, « Sur les ensembles compacts de fonctions sommables », dans Acta Sci. Math., vol. 6, 1933, p. 136–142
Precup, Radu (2002). Methods in nonlinear integral equations. Springer. p. 21. ISBN 978-1-4020-0844-3.