Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if ${\displaystyle E}$ is a normed vector space and ${\displaystyle K}$ is a nonempty convex subset of ${\displaystyle E}$ that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of ${\displaystyle K}$ 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.^[1] Later Namioka and Asplund ^[2] gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.^[3]

Applications

The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups.^[4]

See also

• Fixed-point theorems
• Fixed-point theorems in infinite-dimensional spaces
• Markov-Kakutani fixed-point theorem - abelian semigroup of continuous affine self-maps on compact convex set in a topological vector space has a fixed point