# Markov Kakutani fixed point theorem

Theorem [Markov - Kakutani] - Let $X$ be a topological vector space and $\mathcal{T}$ a commuting family of continuous linear operators in $X$. Suppose $K$ is a compact convex subset of $X$ such that

 $T(K)\subseteq K$

for every $T\in\mathcal{T}$. Then there is a point $x_{0}\in K$ such that $Tx_{0}=x_{0}$ for all $T\in\mathcal{T}$.

Title Markov Kakutani fixed point theorem MarkovKakutaniFixedPointTheorem 2013-03-22 17:33:25 2013-03-22 17:33:25 asteroid (17536) asteroid (17536) 6 asteroid (17536) Theorem msc 54H25 msc 46A99 msc 46A50