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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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    Do you mean *continuous* compactly supported test functions? Because the dual of that space is the space of Radon measures on $\mathbb{R}^n$.2012-11-20
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    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?2012-11-20
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    Those are usually just called "distributions." Here is a good [overview](http://terrytao.wordpress.com/2009/04/19/245c-notes-3-distributions/) of the different varieties of distributions. Hope it helps.2012-11-20
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    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!2012-11-20

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