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nLab > Latest Changes: compactly supported distribution

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2017
    • (edited Oct 18th 2017)

    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).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2017
    • (edited Oct 23rd 2017)

    I have spelled out here the characterization of continuity of a linear map u:C ∞(ℝ n)→ℝu \colon C^\infty(\mathbb{R}^n) \to \mathbb{R} as in Hörmander’s book

    ∃K,k,C(|u(Φ)|≤C∑|α|≤ksupx∈K|∂ αK|) \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 Φ↦supx∈K|∂ αΦ(x)|\Phi \mapsto \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x) \vert}.

  3. 1) semi-norms giving the inductive limit definition of \mathcal{E}(\Omega) should be defined on C^\infty functions, not only on test functions.

    2) typo in the the math font for one name of \mathcal{E}

    lucas

    diff, v21, current

  4. 1) confusion between bump functions and smooth functions

    2) the defined semi-ball (of smooth functions) was in R^n

    lucas

    diff, v21, current

1 to 4 of 4

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