Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if $E$ is a normed vector space and $K$ is a nonempty convex subset of $E$ that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of $K$ 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]

References

^ Ryll-Nardzewski, C. (1962). "Generalized random ergodic theorems and weakly almost periodic functions". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. 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.
^ 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.

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

[1] Ryll-Nardzewski, C. (1962). "Generalized random ergodic theorems and weakly almost periodic functions". Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. 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.

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[2]

[3]

[4]

Applications

See also

References