Definition 2.4.155085BC2

Definition 2.4.1.55085BC2 A distribution on $\Omega $ is a complex linear functional $\chi \mapsto \langle T,\chi \rangle $ on $C_ c^\infty (\Omega )$ such that for each nonempty compact set $K\subset \Omega $ there are $C>0$ and $m\in \mathbb N$ with

A distribution $T$ is real if $\langle T,\chi \rangle $ is real for every real test function $\chi $, and a real distribution is positive if $\langle T,\chi \rangle \geq 0$ for every nonnegative test function $\chi $. The space of distributions on $\Omega $ is denoted by $\mathcal D'(\Omega )$.