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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added statement and proof that compactly supported distributions are equivalently the smooth linear functionals: here
(in the sense of either diffeological spaces, or smooth sets, or formal smooth sets/Cahiers topos).
I have spelled out here the characterization of continuity of a linear map $u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}$ as in Hörmander’s book
$\underset{K,k,C}{\exists} \left( \vert u(\Phi) \vert \;\leq\; C \underset{ {\vert \alpha \vert \leq k} }{\sum} \underset{x \in K}{sup} \vert \partial^\alpha K \vert \right)$from the un-summed seminorms $\Phi \mapsto \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x) \vert}$.
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