Ryll-Nardzewski fixed-point theorem: Difference between revisions

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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 [[Fixed point (mathematics)|fixed]] by each map in the set.)

This theorem was announced by [[Czesław Ryll-Nardzewski]] in <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. Astronom. 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|coauthors=Asplund, E.|title=A geometric proof of Ryll-Nardzewski's fixed point theorem|journal=Bull. Amer. Math. Soc.|volume=73|numberissue=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 journal|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|isbn=2225684103}}</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==