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In the book Distribution theory by Duistermmat/Kolk convergence in the space of test functions is defined as:

$\phi_j \to \phi$ in $C_0^{\infty}(\Omega) $ (where $\Omega$ is an open subset of $\mathbb{R}^n$ )if:

$i)$ There is a compact set $K: \text{supp} \phi_j \subset K$ for every $j$.

$ ii)$ For every multi-index $\alpha$ the sequence $\partial^\alpha \phi_j \to \partial \phi$ uniformly.

I'm not really sure how to interpret this. By "uniformly" in criterion $ii)$ is it meant uniformly on every compact subset of $\Omega$?

It is not a criterion that for every multi-index $\alpha$ there is a compact $K \subset \Omega$ such that $\text{supp} \partial^\alpha \phi_j$? If no is it possible to $\phi_j \to \phi$ in $C_0^{\infty}(\Omega)$ but the support of its derivatives "diverging"?

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    $\begingroup$ Uniformly on $K$ which is the same for each $\alpha$ since it contains the support of each $\phi_j$ (and of the limit). When discarding the compact support condition for the convergence, you obtain the Schwartz space, which is probably easier to understand (it is generated by the translates of $e^{-n x^2}$) but has a smaller dual (the tempered distributions). $\endgroup$ – reuns Jun 20 '17 at 12:19
  • $\begingroup$ Ok, I think I get that, but what about the derivatives? Does each derivative have to "converge in the space of test functions" as well? $\endgroup$ – user202542 Jun 20 '17 at 12:21
  • $\begingroup$ The definition guarantees that $\partial^\alpha\phi_j \to \partial^\alpha\phi$ which is why its dual has those nice properties. $\endgroup$ – reuns Jun 20 '17 at 12:23
  • $\begingroup$ @user202542 I have a silly question, the book you mentioned, is that a good book for distribution theory, or you can suggest better in this matter! $\endgroup$ – MAN-MADE Jun 20 '17 at 12:39
  • $\begingroup$ @MANMAID I've only just started with it but the very first chapters seems good. I'd recommend taking a look in Rudin's Functional Analysis as a complement though. $\endgroup$ – user202542 Jun 25 '17 at 21:13

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