delta distribution

Let U be an open subset of n such that 0U. Then the delta distribution is the mapping

u u(0).

Claim The delta distribution is a distributionPlanetmathPlanetmathPlanetmath of zeroth order, i.e., δ𝒟0(U).

Proof. With obvious notation, we have

δ(u+v) = (u+v)(0)=u(0)+v(0)=δ(u)+δ(v),
δ(αu) = (αu)(0)=αu(0)=αδ(u),

so δ is linear. To see that δ is continuous, we use condition (3) on this this page ( Indeed, if K is a compact set in U, and u𝒟K, then


where |||| is the supremum norm.

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