> [!definition] > > Let $U \subset \real^d$ be [[Open Set|open]] and $F \in \cd'(U)$ be a [[Distribution|distribution]], then there exists a maximal open set $V \subset U$ such that $F|_{\cd(V)} = 0$. The complement of this set is known as the **support** of $F$. > > *Proof*. Let $V$ be the union of all open sets on which $F$ vanishes, then by the [[Gluing Lemma for Distributions]], $F|_{\cd(V)} = 0$. > [!theorem] > > Let $\phi \in C_c^\infty(U)$, then $\supp{\phi} = \supp{T_\phi}$. > > *Proof*. Firstly, $\angles{\phi, \psi} = 0$ for all $\psi \in \cd(\supp{\phi}^c)$, so $\supp{T_\phi} \subset \supp{\phi}$. On the other hand, if $\angles{\phi, \psi} = 0$ for all $\psi \in \cd(\supp{T_\phi}^c)$, then $\phi|_{\supp{T_\phi}^c} = 0$, so $\supp{\phi} \subset \supp{T_\phi}$.