Ryll-Nardzewski fixed-point theorem
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In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E {\displaystyle E} is a normed vector space and K {\displaystyle K} is a nonempty convex subset of E {\displaystyle E} that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of K {\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. 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.
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
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
- written by J. Lurie