1. ## 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 $x$) $(\exists \phi\in {\cal D}(V(x)))$ $\neq 0$.

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

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

Definition of distribution support: Let $\Omega\subseteq \mathbb{R}^n$ 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
$\color{green}\neq 0$.

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.

3. thanks a lot friend!