# Montel’s theorem

Suppose that $G\subset \u2102$ is a region.

###### Theorem (Montel).

A set $\mathrm{F}$ of holomorphic functions^{} $f\mathrm{:}G\mathrm{\to}\mathrm{C}$ is normal (http://planetmath.org/NormalFamily) if and only if $\mathrm{F}$ is
locally bounded.

In other words a sequence of holomorphic functions $\{{f}_{n}\}$ has a subsequence which converges uniformly on compact subsets to a holomorphic function $f:G\to \u2102$ if and only if the set $\{{f}_{n}\}$ is locally bounded.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.

Title | Montel’s theorem |
---|---|

Canonical name | MontelsTheorem |

Date of creation | 2013-03-22 14:17:52 |

Last modified on | 2013-03-22 14:17:52 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30C99 |

Related topic | AscoliArzelaTheorem |

Related topic | SpaceOfAnalyticFunctions |