# closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.

• By the very definition of subspace, yes. – Ataraxia Aug 18 '13 at 18:19
• Finite dimensional normed spaces of the same dimension are isomorphic. A finite subspace of a normed vector space $X$ is thus isomorphic to some $\ell_2^n$. As such, it is complete; thus closed. – David Mitra Aug 18 '13 at 18:29
• @DavidMitra Thanks – chandu1729 Aug 18 '13 at 18:30
• Perhas chandu assumes real scalars? Or at least a complete field for the scalars... otherwise counterexamples exist. – GEdgar Aug 18 '13 at 19:02
• Topologically closed – chandu1729 Aug 18 '13 at 19:36

## 1 Answer

Finite dimensional normed spaces of the same dimension are isomorphic. A finite subspace of a normed vector space X is thus isomorphic to some $\ell_2^n$. As such, it is complete; thus closed.