Support of distribution

Question

How the support of Dirac distribution is $\{0\}$. I have started reading Distributions just a few days before. Can anyone help me out? Thanks in advance.

Answer 1

We say that a distribution $T$ vanishes on an open set $U \subseteq \mathbb{R}$ if for every $\phi \in C_c^\infty(\mathbb{R})$ such that $\operatorname{supp} \phi \subseteq U$ we have $T(\phi) = 0$. The support of a distribution $\operatorname{supp} T$ is defined as the complement of the largest open set on which $T$ vanishes.

Now consider the Dirac distribution $\delta_0$. For the open set $\mathbb{R}\setminus \{0\}$ and $\phi \in C_c^\infty(\mathbb{R})$ such that $\operatorname{supp} \phi \subseteq \mathbb{R}\setminus \{0\}$ we clearly have $\delta_0(\phi) = \phi(0) = 0$. Clearly $\delta_0$ does not vanish on $\mathbb{R}$ so $\mathbb{R}\setminus \{0\}$ is the largest open set which $\delta_0$ vanishes on.

It follows $\operatorname{supp} \delta_0 = \{0\}$.