Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if $$E$$ is a normed vector space and $$K$$ is a nonempty convex subset of $$E$$ that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of $$K$$ has at least one fixed point. (Here, a fixed point of a set of maps is a point that is fixed by each map in the set.)

This theorem was announced by Czesław Ryll-Nardzewski. Later Namioka and Asplund gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.

Applications
The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups.