# 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 DeltaDistribution 2013-03-22 13:45:52 2013-03-22 13:45:52 matte (1858) matte (1858) 6 matte (1858) Definition msc 46-00 msc 46F05 ExampleOfDiracSequence