# bounded

Let $X$ be a subset of $\mathbb{R}$. We say that $X$ is bounded^{} when there exists a real number $M$ such that $$ for all $x\in X$. When $X$ is an interval, we speak of a bounded interval.

This can be generalized first to ${\mathbb{R}}^{n}$. We say that $X\subseteq {\mathbb{R}}^{n}$ is bounded if there is a real number $M$ such that $$ for all $x\in X$ and $\parallel \cdot \parallel $ is the Euclidean distance between $x$ and $y$.

This condition is equivalent^{} to the statement: There is a real number $T$ such that $$ for all $x,y\in X$.

A further generalization^{} to any metric space $V$ says that $X\subseteq V$ is bounded when there is a real number $M$ such that $$ for all $x,y\in X$, where $d$ is the metric on $V$.

Title | bounded |
---|---|

Canonical name | Bounded1 |

Date of creation | 2013-03-22 14:00:00 |

Last modified on | 2013-03-22 14:00:00 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 11 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 54E35 |

Related topic | EuclideanDistance |

Related topic | MetricSpace |

Defines | bounded interval |