Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
[Math] closed subspace of normed vector space
analysisfunctional-analysistopological-vector-spacesvector-spaces
analysisfunctional-analysistopological-vector-spacesvector-spaces
Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
Best 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.