Distributions with support of the form $\left\lbrace x \right\rbrace$

Question

Doing some calculations with Distributions I came up with the following theorem:

THEOREM: Let $O \subseteq \mathbb{R^d}$ be an open subset and $x \in O$. Suppose $T \in \mathcal{D}'(O)$ with $\operatorname{supp}T \subseteq \left\lbrace x \right\rbrace$. Then there exists some $n \in \mathbb{N}_0$ and constants $c_{\alpha} \in \mathbb{C}$ with $|\alpha| \leq n$ such that \begin{align*} T=\sum_{|\alpha| \leq n} c_{\alpha} \partial^{\alpha} \delta_{x}. \end{align*}

Does anybody know where I can find a proof of this?

Answer 1

Hörmander, Analysis of linear PDO's, Theorem 2.3.4.