1. ## About definition of distribution support.

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

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

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

My question,

this def is equivalent of:

$\displaystyle x\in supp(u)$ if:

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

remmark: $\displaystyle {\cal D}(\Omega):=C_0^\infty(\Omega)$

2. Originally Posted by yemino
Hi, I hace a question about the support of a distribution.

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

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

My question,

this def is equivalent of:

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

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

remark: $\displaystyle {\cal D}(\Omega):=C_0^\infty(\Omega)$
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.

3. thanks a lot friend!