# 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 $T{x}_{0}={x}_{0}$ for all $T\in \mathcal{T}$.

Title | Markov Kakutani fixed point theorem |
---|---|

Canonical name | MarkovKakutaniFixedPointTheorem |

Date of creation | 2013-03-22 17:33:25 |

Last modified on | 2013-03-22 17:33:25 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 6 |

Author | asteroid (17536) |

Entry type | Theorem |

Classification | msc 54H25 |

Classification | msc 46A99 |

Classification | msc 46A50 |