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.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
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
- A proof written by J. Lurie
References
Ryll-Nardzewski, C. (1962). "Generalized random ergodic theorems and weakly almost periodic functions". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 10: 271–275. ↩
Namioka, I.; Asplund, E. (1967). "A geometric proof of Ryll-Nardzewski's fixed point theorem". Bull. Amer. Math. Soc. 73 (3): 443–445. doi:10.1090/S0002-9904-1967-11779-8. /wiki/Isaac_Namioka ↩
Ryll-Nardzewski, C. (1967). "On fixed points of semi-groups of endomorphisms of linear spaces". Proc. 5th Berkeley Symp. Probab. Math. Stat. 2: 1. Univ. California Press: 55–61. ↩
Bourbaki, N. (1981). Espaces vectoriels topologiques. Chapitres 1 à 5. Éléments de mathématique. (New ed.). Paris: Masson. ISBN 2-225-68410-3. 2-225-68410-3 ↩