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, we use 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. $\Box$

Title	delta distribution
Canonical name	DeltaDistribution
Date of creation	2013-03-22 13:45:52
Last modified on	2013-03-22 13:45:52
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	6
Author	matte (1858)
Entry type	Definition
Classification	msc 46-00
Classification	msc 46F05
Related topic	ExampleOfDiracSequence