# $\mathbb{R}^{n}$ 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 mathbbRnIsNotACountableUnionOfProperVectorSubspaces 2013-03-22 14:59:03 2013-03-22 14:59:03 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Result msc 54E52