Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if formula_1 is a normed vector space and formula_2 is a nonempty convex subset of formula_1 that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of formula_2 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.