3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ why do you think $C^\infty$ is normable? $\endgroup$ – oldrinb Sep 15 '13 at 2:46
  • $\begingroup$ 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 $C^\infty$ $\endgroup$ – Twiltie Sep 15 '13 at 2:49
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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) $\endgroup$ – Twiltie Sep 15 '13 at 2:53
  • 1
    $\begingroup$ @Twiltie: Dear Twiltie, Books on locally convex vector spaces (e.g. Bourbaki's Topological vector spaces, or the shortish CUP book by Robinson and Robinson with a similar name) will discuss this example, but perhaps surrounded by more foundational abstraction than you might want. Books on distributions (probably more old-fashioned ones, at least) will also discuss it. One basic thing to remember is that to guarantee that a uniform limit of differentiable functions is again differentiable, you need to assume that the derivatives converge as well. (This is a standard theorem form real ... $\endgroup$ – Matt E Sep 15 '13 at 2:57
  • 1
    $\begingroup$ ... 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, $\endgroup$ – Matt E Sep 15 '13 at 2:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.