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 ${\displaystyle B}$ 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 ${\displaystyle |a|<\delta .}$

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.