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I'm reading "Pseudo-differential Operators and the Nash-Moser Theorem" and at the top of on page 8 they write: "Finally, we note that if $u \in C^0(\Omega)$, then the support of $u$ defined above coincides with the closure of $\{x \in \Omega \mid u(x) \neq 0 \}$." $\Omega$ here is an open subset of $\mathbb{R}^n$.

The support of a distribution $u$ was defined on the previous page as "the complement in $\Omega$ of the points in the neighbourhood of which $u$ is zero".

Question 1: What's the difference between $C(\Omega)$ and $C^0(\Omega)$?

Question 2: From $\operatorname{supp}{u} \subset \Omega$ I deduce that we think of $u$ here as a distribution on a subset of reals. How does this make sense? If my test functions are constants then the integral of $cu$ might not be finite.

Thanks for your help.

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    Test functions are usually defined as smooth with compact support, so the integral exists.2012-02-23
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    Yes I know. But if the test functions are a subset of $\mathbb{R}$ then that's false.2012-02-23
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    Can you give an example? (Also, I don't have my copy of that book handy, but doesn't it have a table of notations near the end?)2012-02-23
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    @WillieWong Yes it does have a table but $C^0$ is not on it. An example of what?2012-02-23
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    An example of a distribution represented by a function $u$ such that its support is contained in some open set $\Omega$ and such that its integral against a test function is not finite.2012-02-23

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