Definition:Support of Distribution

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Let $\Omega \subseteq \R^n$ be an open set.

Let $\map \DD \Omega$ be the space of continuous functions compactly supported in $\Omega$.

Let $\map {\DD'} \Omega$ be the distribution space.

Let $T \in \map {\DD'} \Omega$ be a distribution.

The support $\map {\mathrm {supp} } T \subseteq \Omega$ of $T$ is defined by:

$\ds x \notin \map {\mathrm {supp} } T$ if and only if:
there exists an open neighborhood $U$ of $x$ such that:
for all $\phi \in \map \DD \Omega$ such that $\map {\mathrm {supp} } \phi \subseteq U$:
$\map T \phi = 0$