# ${\mathbb{R}}^{n}$ is not a countable union of proper vector subspaces

${\mathbb{R}}^{n}$ is not a countable^{} union of proper vector subspaces.

Proof

We know that every finite dimensional proper subspace of a normed space is nowhere dense. Besides, ${\mathbb{R}}^{n}$ is a Banach space^{}, so the results follows directly.

Title | ${\mathbb{R}}^{n}$ is not a countable union of proper vector subspaces |
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Canonical name | mathbbRnIsNotACountableUnionOfProperVectorSubspaces |

Date of creation | 2013-03-22 14:59:03 |

Last modified on | 2013-03-22 14:59:03 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Result |

Classification | msc 54E52 |