# uniform convergence

Let $X$ be any set, and let $(Y,d)$ be a metric space.
A sequence ${f}_{1},{f}_{2},\mathrm{\dots}$ of functions mapping $X$ to $Y$ is said to be
*uniformly convergent* to another function $f$ if, for each $\epsilon >0$, there exists $N$ such that, for all $x$ and all $n>N$, we have $$.
This is denoted by ${f}_{n}\stackrel{\mathit{u}}{\to}f$, or “${f}_{n}\to f$ uniformly” or, less frequently, by ${f}_{n}\rightrightarrows f$.

Title | uniform convergence |
---|---|

Canonical name | UniformConvergence |

Date of creation | 2013-03-22 13:13:49 |

Last modified on | 2013-03-22 13:13:49 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 14 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 40A30 |

Related topic | CompactOpenTopology |

Related topic | ConvergesUniformly |

Defines | uniformly convergent |