Definition:Support of Distribution

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