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In [[functional analysis]], 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> which is [[compact space|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
▲In [[functional analysis]], 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> which is [[compact space|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 a [[Fixed point (mathematics)|fixed point]] for each of the set's members.)
This theorem was announced by [[Czesław Ryll-Nardzewski]] in <ref>{{cite article|first=C.|last=Ryll-Nardzewski|title=Generalized random ergodic theorems and weakly almost periodic functions|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.|volume=10|year=1962|pages=271-275}}</ref>. Later Namioka and Asplund <ref>{{cite article|first=I.|last=Namioka|coauthors=Asplund, E.|title=A geometric proof of Ryll-Nardzewski's fixed point theorem|journal=Bull. Amer. Math. Soc.|volume=73|number=3|year=1967|pages=443-445}}</ref> gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit in <ref>{{cite article|first=C.|last=Ryll-Nardzewski|title=On fixed points of semi-groups of endomorphisms of linear spaces|journal=Proc. 5-th 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 edition|publisher=Masson|___location=Paris|year=1981}}</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 -->
==See also==
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==References==
<references />
* Andrzej Granas and James Dugundji, ''Fixed Point Theory'' (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
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