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I use $C_0(\Omega)$ and $C_c(\Omega)$ pretty much interchangeably but I'm trying to be more precise now so I'd like to know what is the difference between these spaces?

$C_0$ is commonly referred to as the set of all continuous functions on $\Omega \in \mathbb{R}^n$ that vanish at infinity (or on the boundary).

$C_c(\Omega)$ is the space of all continuous functions on $\Omega$ with compact support.

It holds that $C_c(\Omega) \subset C_0(\Omega)$. So what functions are in $C_0(\Omega)$ that are not in $C_c(\Omega)$?

ManUtdBloke
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  • $C_c(\Omega) \subset C_0(\Omega)$, of course. – copper.hat Jun 30 '18 at 20:13
  • @copper.hat I'm not sure that this is really a duplicate - the linked one is about $\Omega = \mathbb{R}$ and vanishing at infinity, whereas this one is far more general. If anything, the dupe should go the other way IMHO. –  Jun 30 '18 at 20:20
  • @T.Bongers: Other than the $n>1$ part, or an elaboration of the characteristics of $\Omega$, I don't see a big distinction? But I am open to reason (mostly) - can I undup? – copper.hat Jun 30 '18 at 20:28
  • @copper.hat It's up to you, but there are actually quite a few other duplicates it seems - https://math.stackexchange.com/questions/1561560/relationship-among-the-function-spaces-c-c-infty-omega-c-c-infty-overli?rq=1 or https://math.stackexchange.com/questions/69262/clc-c-omega-is-a-subset-of-c-0-omega?rq=1 already have quite a bit of overlap. So never mind. –  Jun 30 '18 at 20:35
  • @T.Bongers: Unfortunately my reaction to a question is often a function of the questions have encountered just before... – copper.hat Jun 30 '18 at 20:37
  • Note that in fields like geometric analysis one needs to be more careful with the statement $C_c(Ω)⊂C_0(Ω)$ since compact domains (compact manifolds-with-boundary $\Omega = M$) are sometimes also admitted. In this case, a compactly supported function need not vanish on the boundary (unless, of course, it is compactly supported in the interior of $\mathrm{int}{M} = M \setminus \partial M$ or we define $C_c(M)$ that way). – balu Jan 22 '20 at 15:31
  • (continued) It is probably for this reason that I've seen many people use $C^k_0(M)$ for functions which have compact support in the manifold(-with-boundary) $M$ and that vanish on the boundary (including its derivatives up to $k$-th order, while $C^k_c(M) \supset C^k_0(M)$ then simply denotes those functions with compact support in $M$. This definition of $C^k_0(M)$ becomes particularly useful when frequently using Stokes' theorem (which is your bread and butter in geometric analysis) without wanting to consider boundary terms. – balu Jan 22 '20 at 16:42
  • (continued) Especially so when $M$ might not be a compact manifold (let alone a Cauchy-complete Riemannian manifold), in which case "vanishing along the (manifold) boundary" would not be enough for Stokes' theorem to hold. – balu Jan 22 '20 at 16:42

2 Answers2

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Remember that $\partial \Omega$ is a closed set. So if $f \in C_c(\Omega)$ has compact support in $\Omega$, the support of $f$ and $\partial \Omega$ have a strictly positive distance from each other. That means that you can fit a little open set between $\Omega$ and the support of $f$.

This is a much stronger condition than simply requiring $f$ tends to zero on the boundary. Being able to have a constraint on the distance between $\Omega^c$ and where $f$ has interesting behavior is frequently important in proving things about $f$.

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For example you can consider $f(x)=e^{-||x||}$

Federico Fallucca
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