Fréchet–Kolmogorov theorem

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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 Lp space. It can be thought of as an Lp 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 ${\displaystyle B}$ be a bounded set in ${\displaystyle L^{p}(\mathbb {R} ^{n})}$, with ${\displaystyle p\in [1,\infty )}$.

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

1. ${\displaystyle \lim _{r\to \infty }\int _{|x|>r}\left|f\right|^{p}=0}$ uniformly on B,
2. ${\displaystyle \lim _{a\to 0}\Vert \tau _{a}f-f\Vert _{L^{p}(\mathbb {R} ^{n})}=0}$ uniformly on B,

where ${\displaystyle \tau _{a}f}$ denotes the translation of ${\displaystyle f}$ by ${\displaystyle a}$, that is, ${\displaystyle \tau _{a}f(x)=f(x-a).}$

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