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delta distribution

(Definition)

Let 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., $\delta\in \mathcal{D}'^0(U)$ .

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. Indeed, if

, and $u\in \mathcal{D}_K$ , then

$\displaystyle \vert\delta(u)\vert = \vert u(0)\vert \le \vert\vert u\vert\vert _\infty,$

where $\vert\vert\cdot\vert\vert _\infty$ is the supremum norm. $\Box$

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Cross-references: supremum norm, compact set, continuous, obvious, order, distribution, mapping, open subset
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This is version 3 of delta distribution, born on 2003-07-17, modified 2006-01-16.
Object id is 4468, canonical name is DeltaDistribution.
Accessed 4310 times total.

Classification:

AMS MSC:	46F05 (Functional analysis :: Distributions, generalized functions, distribution spaces :: Topological linear spaces of test functions, distributions and ultradistributions)
	46-00 (Functional analysis :: General reference works )

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