delta distribution

Let $U$ be an open subset of $\\mathbb{R}^{n}$ such that $0\\in U$ .Then the delta distribution is the mapping

	$\\displaystyle\\delta:\\mathcal{D}(U)$	$\\displaystyle\\to$	$\\displaystyle\\mathbb{C}$
	$\\displaystyle u$	$\\displaystyle\\mapsto$	$\\displaystyle u(0).$

Claim The delta distribution is a distribution of zeroth order, i.e., $\\delta\\in\\mathcal{D}^{\\prime 0}(U)$ .

Proof. With obvious notation, we have

	$\\displaystyle\\delta(u+v)$	$\\displaystyle=$	$\\displaystyle(u+v)(0)=u(0)+v(0)=\\delta(u)+\\delta(v),$
	$\\displaystyle\\delta(\\alpha u)$	$\\displaystyle=$	$\\displaystyle(\\alpha u)(0)=\\alpha u(0)=\\alpha\\delta(u),$

so $\\delta$ is linear. To see that $\\delta$ is continuous, weuse condition (3) on this this page (http://planetmath.org/Distribution4).Indeed, if $K$ is a compact set in $U$ , and $u\\in\\mathcal{D}_{K}$ , then

|\\delta(u)|=|u(0)|\\leq||u||_{\\infty},

where $||\\cdot||_{\\infty}$ is the supremum norm. $\\Box$