Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am reading something on Sobolev spaces.

We define $D(\Omega)$ to be the set of function in $C^{\infty}(\Omega)$ that has compact support in $\Omega$.

I know that compact support is completion of the set where a function is non-zero.

but, what does it mean by "have compact support in $\Omega$"? Does it mean it is a function such that its compact support is a subset of $\Omega$?

share|cite|improve this question
Yes, that's exactly what it means. These are functions which are only nonzero on (pre)compact subsets of $\Omega$. – Neal Oct 16 '12 at 16:09
up vote 1 down vote accepted

I'll turn my comment into an answer. If $\Omega\subset\mathbb{R}^n$ is a domain, a function $f$ on $\mathbb{R}^n$ "has compact support in $\Omega$" provided $supp(f)$ is a compact subset of $\Omega$. Functions with compact support in $\Omega$ are those which are only nonzero on precompact subsets of $\Omega$.

share|cite|improve this answer

You can define support of a function by $$supp(f)=\overline{\{x\in\Omega: f(x)\neq 0\}}$$

Now when you say that the function have compact support, you mean that the set $supp(f)$ is compact.

share|cite|improve this answer
Maybe I did not make this clear enough, I wanted to know what it meant by have "compact support in $\Omega$". The "in $\Omega$" bit is the key. It has been answered already. Don't worry – Lost1 Oct 16 '12 at 18:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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