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

References

1. 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.
2. 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.
3. 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.
4. Bourbaki, N. (1981). Espaces vectoriels topologiques. Chapitres 1 à 5. Éléments de mathématique. (New ed.). Paris: Masson. ISBN 2-225-68410-3.

• Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
• A proof written by J. Lurie