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

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?

• why do you think $C^\infty$ is normable? – oldrinb Sep 15 '13 at 2:46
• Well to be honest I don't really know to much about norms, so I hadn't even thought of the possibility that you can't place a norm on $C^\infty$ – Twiltie Sep 15 '13 at 2:49

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.
• Thank you! I didn't realize that you couldn't put a single norm on $C^\infty$ but the seminorm idea makes sense (It seems I need to read some more) – Twiltie Sep 15 '13 at 2:53
• ... analysis books.) So to get a nice topological structure on $C^{\infty}$, you are more-or-less forced to use the whole collection of seminorms $s_n$. Cheers, – Matt E Sep 15 '13 at 2:57