delta distributionDeltaDistribution2003-07-17 07:24:382006-01-16 18:17:24Definitiondistribution% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}\newcommand{\cD}[0]{\mathcal{D}}
Let $U$ be an open subset of $\sR^n$ such that $0\in U$.
Then the \emph{delta distribution} is the mapping
\begin{eqnarray*}
\delta : \cD(U) &\to & \sC \\
u &\mapsto & u(0).
\end{eqnarray*}
{\bf Claim} The delta distribution is a distribution of zeroth order, i.e.,
$\delta\in \cD'^0(U)$.
\emph{Proof.} With obvious notation, we have
\begin{eqnarray*}
\delta(u+v)&=&(u+v)(0)=u(0)+v(0) = \delta(u) + \delta(v),\\
\delta(\alpha u) &=& (\alpha u)(0)=\alpha u(0)=\alpha \delta(u),
\end{eqnarray*}
so $\delta$ is linear. To see that $\delta$ is continuous, we
use condition (3) on this \PMlinkname{this page}{Distribution4}.
Indeed, if $K$ is a compact set in $U$, and $u\in \cD_K$, then
$$ |\delta(u)| = |u(0)| \le ||u||_\infty,$$
where $||\cdot||_\infty$ is the supremum norm. $\Box$