Ryll-Nardzewski fixed-point theorem

The Ryll-Nardzewski fixed point theorem states that if $E$ is a normed vector space and $K$ is nonempty convex subset of $E$, which is closed for the weak topology, then every group of affine isometries of $K$ has a fixed point.

Applications

The Ryll-Nardzewski theorem yields the existence of a Haar measure on (locally ?) compact groups.

References

• Nicolas Bourbaki, Éléments de mathématique - Espaces vectoriels topologiques, Hermann (1964).