Why is the subspace of polynomials with degree $\leq$ n nowhere dense in $\mathbb{R}[X]$?

Question

There's a popular application of Baire's Category Theorem that shows that $\mathbb{R}[X]$ (the space of all polynomials with real coefficients) is not complete by showing it is a countable union of nowhere dense sets. The sets chosen are all polynomials of degree less than or equal to n.

My question is - why are these sets nowhere dense in any given norm? I can see why this is the case for $l_p$ norms - but why is this generally true for $any$ norm?

Thanks!

Answer 1

This is true, because your subspaces are finite dimensional subspaces. Finite dimensional subspaces are closed with respect to any norm (basically, because all norms on finite dimensional spaces are equivalent, so the finite dimensional subspace will be complete, hence closed).

So the claim left is that the subspace has nonempty interior.

But it is easy to see that any subspace with nonempty interior has to be the whole vector space (take a small $\varepsilon$-Ball around $0$ which is contained in $V$ by assumption and "inflate" that).

EDIT: BTW, this shows that NO vector space with $\dim(V) = \infty$ can be endowed with a norm that makes it complete as long as it admits a countable basis.