Ryll-Nardzewski fixed-point theorem
In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if is a normed vector space and is a nonempty convex subset of that is compact under the weak topology, then every group of affine isometries of has at least one fixed point.
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.
