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Is there a generally accepted name for the space of distributions over smooth compactly supported test functions $\mathcal{D}(\mathbb{R}^n)$?

I know that distributions over the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ are called "tempered distributions," and distributions over $C^\infty(\mathbb{R}^n)$ test functions are called "distributions with compact support."

I can't seem to find a name for the space $\mathcal{D}'(\mathbb{R}^n)$ mentioned anywhere.

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    $\begingroup$ Do you mean continuous compactly supported test functions? Because the dual of that space is the space of Radon measures on $\mathbb{R}^n$. $\endgroup$
    – icurays1
    Nov 20, 2012 at 5:16
  • $\begingroup$ Sorry, should have said $\mathcal{C}^\infty$ test functions with compact support. Is there a name for the space of distributions defined on those test functions? $\endgroup$
    – B0112358
    Nov 20, 2012 at 5:56
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    $\begingroup$ Those are usually just called "distributions." Here is a good overview of the different varieties of distributions. Hope it helps. $\endgroup$
    – icurays1
    Nov 20, 2012 at 6:10
  • $\begingroup$ Ah ok I get it. I forgot that tempered distributions and compactly supported Distributions are a subset of $\mathcal{D}'(\Omega)$. Thanks for the blog link too very helpful! $\endgroup$
    – B0112358
    Nov 20, 2012 at 17:53

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This question has been answered in comments:

Those are usually just called "distributions." Here is a good overview of the different varieties of distributions. Hope it helps. – icurays1 Nov 20 '12 at 6:10

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