nForum - Discussion Feed (compactly supported distribution) 2021-07-26T16:57:21-04:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "compactly supported distribution" (66085) https://nforum.ncatlab.org/discussion/8091/?Focus=66085#Comment_66085 2017-10-23T09:39:26-04:00 2021-07-26T16:57:21-04:00 Urs https://nforum.ncatlab.org/account/4/ I have spelled out here the characterization of continuity of a linear map u&colon;C &infin;(&Ropf; n)&rightarrow;&Ropf;u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R} as in ...

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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Urs comments on "compactly supported distribution" (65985) https://nforum.ncatlab.org/discussion/8091/?Focus=65985#Comment_65985 2017-10-18T08:24:16-04:00 2021-07-26T16:57:21-04:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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).

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