Definition:Zero Mapping/Distribution

Let $\map \DD \R$ be the test function space.
Let $\mathbf 0 \in \map {\DD'} \R$ be a distribution.
$\forall \phi \in \map \DD \R : \map {\mathbf 0} \phi = 0$
Then $\mathbf 0$ is referred to as the zero distribution.