Ryll-Nardzewski fixed-point theorem: Difference between revisions

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TheIn [[functional analysis]], a branch of mathematics, the '''Ryll-Nardzewski fixed -point theorem''' states that if <math>E</math> is a [[normed vector space]] and <math>K</math> is a nonempty [[Convex set|convex]] subset of <math>E</math>, whichthat is closed[[compact forspace|compact]] under the [[weak topology]], then every [[group (mathematics)|group]] (or equivalently: every [[semigroup]]) of [[affine map|affine]] [[isometry|isometries]] of <math>K</math> has at least one fixed point. (Here, a ''fixed point'' of a set of maps is a point that is [[fixedFixed point (mathematics)|fixed]] by each map in the set.)

This theorem was announced by [[Czesław Ryll-Nardzewski]].<ref>{{cite journal|first=C.|last=Ryll-Nardzewski|title=Generalized random ergodic theorems and weakly almost periodic functions|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.|volume=10|year=1962|pages=271–275}}</ref> Later Namioka and Asplund <ref>{{cite journal|doi=10.1090/S0002-9904-1967-11779-8|first=I.|last=Namioka|author1-link= Isaac Namioka |author2=Asplund, E. |title=A geometric proof of Ryll-Nardzewski's fixed point theorem|journal=Bull. Amer. Math. Soc.|volume=73|issue=3|year=1967|pages=443–445|doi-access=free}}</ref> gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.<ref>{{cite journal|first=C.|last=Ryll-Nardzewski|title=On fixed points of semi-groups of endomorphisms of linear spaces|journal=Proc. 5th Berkeley Symp. Probab. Math. Stat|volume=2: 1|publisher=Univ. California Press|year=1967|pages=55–61}}</ref>

==Applications==

The Ryll-Nardzewski theorem yields the existence of a [[Haar measure]] on compact groups.<ref>{{cite book|first=N.|last=Bourbaki|title=Espaces vectoriels topologiques. Chapitres 1 à 5|series=Éléments de mathématique.|edition=New|publisher=Masson|___location=Paris|year=1981|isbn=2-225-68410-3}}</ref><!-- there seems to be a gap in the approach described in Conway's "A Course in Functional Analysis", about the treatment of weak vs. weak* topology -->

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

==See also==

* [[Fixed-point theorem]]s

* [[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==

<references />

* [[Nicolas Bourbaki]], ''Éléments de mathématique - Espaces vectoriels topologiques'', Hermann (1964).

* Andrzej Granas and [[James Dugundji]], ''Fixed Point Theory'' (2003) Springer-Verlag, New York, {{isbn|0-387-00173-5}}.

* [http://www.math.harvard.edu/~lurie/261ynotes/lecture26.pdf A proof written by J. Lurie]

{{Functional analysis}}

[[Category:Fixed-point pointstheorems]]

[[Category:FunctionalTheorems in functional analysis]]

[[Category:Mathematical theorems]]