Ryll-Nardzewski fixed-point theorem

Ryll-Nardzewski fixed-point theorem In functional analysis, ein Zweig der Mathematik, the Ryll-Nardzewski fixed-point theorem states that if {Anzeigestil E} is a normed vector space and {Anzeigestil K} is a nonempty convex subset of {Anzeigestil E} that is compact under the weak topology, then every group (oder gleichwertig: every semigroup) of affine isometries of {Anzeigestil K} has at least one fixed point. (Hier, 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 References ^ Ryll-Nardzewski, C. (1962). "Generalized random ergodic theorems and weakly almost periodic functions". Stier. Akad. Polieren. Wissenschaft. Privatgelände. Wissenschaft. Mathematik. Astron. Phys. 10: 271–275. ^ Namioka, ICH.; Asplund, E. (1967). "A geometric proof of Ryll-Nardzewski's fixed point theorem". Stier. Amer. Mathematik. 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". Proz. 5th Berkeley Symp. Probab. Mathematik. Stat. Univ. California Press. 2: 1: 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 hide vte Functional analysis (Themen – Glossar) Leerzeichen BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraisch/topologisch)lokal konvexreflexivseparable Theoreme Hahn-BanachRiesz-Darstellunggeschlossener Graphgleichmäßiges BeschränktheitsprinzipKakutani-FixpunktKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operatoren adjointboundcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebren Banach-AlgebraC*-AlgebraSpektrum einer C*-AlgebraOperator-Algebravon Gruppenalgebra einer lokalvariant-kompakten Gruppe SubraumproblemMahlersche Vermutung Anwendungen Hardy-RaumSpektraltheorie gewöhnlicher DifferentialgleichungenWärmekernindexsatzVariationsrechnungFunktionsrechnungIntegraloperatorJones-PolynomTopologische QuantenfeldtheorieNichtkommutative GeometrieRiemann-HypotheseVerteilung (oder verallgemeinerte Funktionen) Fortgeschrittene Themen Approximation PropertyBalanced SetChoquet-TheorieSchwache TopologieBanach-Mazur-AbstandTomita-Takesaki-Theorie Kategorien: Fixed-point theoremsTheorems in functional analysis