Ryll-Nardzewski fixed-point theorem
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In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if is a normed vector space and
is a nonempty convex subset of
that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of
has at least one fixed point. (Here, 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]