What norm makes $C^\infty[a,b]$ a complete space?

Question

I have been searching for some common norms used on vector spaces of functions but I am not having any luck finding what the most common norm is to use on $C^\infty[a,b]$ More specifically I would like to find a norm that makes that space complete. Does such a norm exist?

Also what are some good resources that deal with spaces of functions, preferably at an advanced undergraduate or beginner graduate level?

Answer 1

Typically you would make it a Frechet space, by using the countable collection of seminorms given by taking the supremum of each derivative.

I.e. define $s_n(f)$ to be the sup of $f^{(n)}$ on $[a,b]$, and consider the weakest topology for which all the $s_n$ are continuous. This makes $C^{\infty}[a,b]$ a Frechet space.

This is the natural topology to use, in that a sequence of functions converges iff the sequence of $n$th derivatives converges uniformly for each $n$. As far as I know it can't be described by a single norm.