# Interpreting a problem on Frechet spaces (topology)

1. Dec 11, 2011

### Mathmos6

1. The problem statement, all variables and given/known data
I've been given the following problem:

"Suppose that U is a finite-dimensional subspace of a Fréchet space $(V,\tau)$. Show that the subspace topology on U is the usual topology (given for example by a Euclidean norm) and that U is a closed linear subspace of V."

I feel a bit confused: I'm not sure what precisely is meant by 'the usual topology (given for example by a Euclidean norm)', but IIRC any 2 norms on a finite dimensional vector space are equivalent (to e.g. the Euclidean norm) so am I right in thinking that the question just requires me to show that the subspace topology on U is a topology induced by some norm (and therefore equivalent to any norm on U)?

I suppose to show closure is fairly simple, just play around with bases and limits, and it's a subspace of a vector space so linearity is also trivial, right? So then it's only the first part which I need help with, if I understand everything else correctly - thanks!