# 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 be a bounded set in , with .

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

- uniformly on
*B*, - uniformly on
*B*,

where denotes the translation of by , that is,

The second property can be stated as such that with

## 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.

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