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Quick question regarding the generally accepted definition for a space.

Suppose we consider the space $C_{0}^{\infty}(\Omega)$ of smooth functions of with compact support in $\Omega$, do we generally require strict containment of the support in $\Omega$, or do we allow the support to coincide with $\Omega$ in the case of a compact set $\Omega$ and the space to be equivalent to $C^{\infty} (\Omega)$.

guest38234
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  • I think that you allow the equality when it is possible. – D.L. Sep 07 '15 at 19:40
  • But for your last sentence you don't only need $\Omega$ closed ($\mathbb R^n$ is closed...), you need really $\Omega$ compact. – D.L. Sep 07 '15 at 19:42
  • Yes, you are indeed correct regarding the last sentence. I will update. – guest38234 Sep 07 '15 at 19:44
  • So for your question, the answer is no, we do not require a strict containment, the only thing required is that ${x | f(x)\neq 0}$'s closure is compact. – D.L. Sep 07 '15 at 19:46
  • Could you perhaps point me to a reference that notes this fact? I want to be sure I am using the notation correctly when defining the distributional derivative. It would seem that the definition of the weak derivative will loose its meaning if we use $C^{\infty}_{0}(\Omega)$ where $\Omega$ is compact. – guest38234 Sep 07 '15 at 19:47
  • Ok, let me two minutes :) – D.L. Sep 07 '15 at 19:47
  • http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/Real-and-Complex-Analysis-by-Walter-Rudin.pdf

    Page 38.

    The notation $C_c$ is your $C_0$ (in fact $C_0$, is used when $\Omega$ is an open subset of $\mathbb R^n$ for my experience)

     – D.L. Sep 07 '15 at 19:51
  • What makes you think one would exclude the posibility that the support coincides with $\Omega$? I'd expect in general that smooth functions with compact support means precisely that. – skyking Sep 07 '15 at 19:55
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    These spaces seem to generally be used in cases where we require the property that they are 0 within some neighborhood the boundary of the domain. In the case of a closed set this property is lost and hence, before I note this in the report I am writing, I wanted to ensure that my understanding was correct.

    It would seem that in this case, distributional derivatives over a closed set should therefore make use of smooth functions with compact support over the appropriate open set instead.

     – guest38234 Sep 07 '15 at 19:59

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The support of a function is closed, by definition.

Every closed subset of a compact set is compact.

So, if $\Omega$ is a compact set, the subscript is pointless. Just write $C^\infty (\Omega)$ then. Better yet, write $C^\infty(K)$ because the letter $K$ is typically associated with compact sets, while $\Omega$ is associated with open sets.

Generally, to work with notions of smoothness in a closed set, one assumes the existence of a larger open set on which smoothness holds.