# Hilbert parallelotope

The *Hilbert parallelotope* ${I}^{\omega}$ is a closed subset of the Hilbert space^{} $\mathbb{R}{\mathrm{\ell}}^{2}$ (The symbol ’$\mathbb{R}$’ has been prefixed to indicate that the field of scalars is $\mathbb{R}$.) defined as

$${I}^{\omega}=\{({a}_{0},{a}_{1},{a}_{2},\mathrm{\dots})\mid 0\le {a}_{i}\le 1/(i+1)\}$$ |

As a topological space^{}, ${I}^{\omega}$ is homeomorphic to the product^{} of a countably infinite^{} number of copies of the closed interval^{} $[0,1]$. By Tychonoff^{}’s theorem^{}, this product is compact^{}, so the Hilbert parallelotope is a compact subset of Hilbert space. This fact also explains the notation ${I}^{\omega}$.

The Hilbert parallelotope enjoys a remarkable universality property — every second countable metric space is homeomorphic to a subset of the Hilbert parallelotope. Since second countability is hereditary, the converse^{} is also true — every subset of the Hilbert parallelotope is a second countable metric space.

Title | Hilbert parallelotope |
---|---|

Canonical name | HilbertParallelotope |

Date of creation | 2013-03-22 14:38:32 |

Last modified on | 2013-03-22 14:38:32 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 46C05 |

Synonym | Hilbert cube |