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. The theorem is named after Maurice René Fréchet and Andrey Kolmogorov.

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:

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

• Marcel Riesz, « Sur les ensembles compacts de fonctions sommables », dans Acta Sci. Math., vol. 6, 1933, p. 136–142