# equibounded

Let $X$ and $Y$ be metric spaces. A family $F$ of functions from $X$ to $Y$ is said to be *equibounded* if there exists a bounded^{} subset $B$ of $Y$ such that for all $f\in F$ and all $x\in X$ it holds $f(x)\in B$.

Notice that if $F\subset {\mathcal{C}}_{b}(X,Y)$ (continuous bounded functions) then $F$ is equibounded if and only if $F$ is bounded (with respect to the metric of uniform convergence^{}).

Title | equibounded |
---|---|

Canonical name | Equibounded |

Date of creation | 2013-03-22 13:16:55 |

Last modified on | 2013-03-22 13:16:55 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 5 |

Author | paolini (1187) |

Entry type | Definition |

Classification | msc 54E35 |

Synonym | equi bounded |

Synonym | equi-bounded |