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