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Math Help - About definition of distribution support.

  1. #1
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    About definition of distribution support.

    Hi, I hace a question about the support of a distribution.

    Definition of distribution support: Let \Omega\subseteq \mathbb{R}^n be an open set, and u\in {\cal D}'(\Omega ). the support of u is the complement of
    the set

    \{ x : \;\; u=0 \;\;\textrm{on a neighbourhood of}\;\; x\}.

    My question,

    this def is equivalent of:

    x\in supp(u) if:

    x\in\Omega is such that (\forall V(x) neighbourhood of x) (\exists \phi\in {\cal D}(V(x))) <u,\phi>\neq 0.


    remmark: {\cal D}(\Omega):=C_0^\infty(\Omega)
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  2. #2
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    Quote Originally Posted by yemino View Post
    Hi, I hace a question about the support of a distribution.

    Definition of distribution support: Let \Omega\subseteq \mathbb{R}^n be an open set, and u\in {\cal D}'(\Omega ). the support of u is the complement of
    the set

    \color{red}\{ x : \;\; u=0 \;\;\textrm{on a neighbourhood of}\;\; x\}.

    My question,

    this def is equivalent of:

    \color{green}x\in supp(u) if:

    \color{green}x\in\Omega is such that \color{green}(\forall V(x) neighbourhood of \color{green}x\ (\exists \phi\in {\cal D}(V(x))) such that
    \color{green}<u,\phi>\neq 0.


    remark: {\cal D}(\Omega):=C_0^\infty(\Omega)
    I don't see that the set in red makes sense at all. A distribution is not necessarily a function, and I do not see how to attach a meaning to the statement "u = 0 on a neighbourhood of x". In fact, a distribution is defined in terms of its action on test functions, so I would take the statement in green as the only sensible definition of the support of u.
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  3. #3
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    thanks a lot friend!
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