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In [[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> 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]].<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|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}}</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==
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