Editing Distribution (mathematics) (section)

===Gluing and distributions that vanish in a set===

{{Math theorem
| name = Theorem{{sfn |Trèves|2006|pp=253-255}}
| math_statement = Let <math>(U_i)_{i \in I}</math> be a collection of open subsets of <math>\R^n.</math> For each <math>i \in I,</math> let <math>T_i \in \mathcal{D}'(U_i)</math> and suppose that for all <math>i, j \in I,</math> the restriction of <math>T_i</math> to <math>U_i \cap U_j</math> is equal to the restriction of <math>T_j</math> to <math>U_i \cap U_j</math> (note that both restrictions are elements of <math>\mathcal{D}'(U_i \cap U_j)</math>). Then there exists a unique <math display=inline>T \in \mathcal{D}'(\bigcup_{i \in I} U_i)</math> such that for all <math>i \in I,</math> the restriction of {{mvar|T}} to <math>U_i</math> is equal to <math>T_i.</math>
}}

Let {{mvar|V}} be an open subset of {{mvar|U}}. <math>T \in \mathcal{D}'(U)</math> is said to '''{{em|vanish in {{mvar|V}}}}''' if for all <math>f \in \mathcal{D}(U)</math> such that <math>\operatorname{supp}(f) \subseteq V</math> we have <math>Tf = 0.</math> {{mvar|T}} vanishes in {{mvar|V}} if and only if the restriction of {{mvar|T}} to {{mvar|V}} is equal to 0, or equivalently, if and only if {{mvar|T}} lies in the [[kernel (algebra)|kernel]] of the restriction map <math>\rho_{VU}.</math>

{{Math theorem
| name = Corollary{{sfn |Trèves|2006| pp=253-255}}
| math_statement = Let <math>(U_i)_{i \in I}</math> be a collection of open subsets of <math>\R^n</math> and let <math display=inline>T \in \mathcal{D}'(\bigcup_{i \in I} U_i).</math> <math>T = 0</math> if and only if for each <math>i \in I,</math> the restriction of {{mvar|T}} to <math>U_i</math> is equal to 0.
}}

{{Math theorem| name=Corollary{{sfn |Trèves|2006|pp=253-255}}| math_statement= The union of all open subsets of {{mvar|U}} in which a distribution {{mvar|T}} vanishes is an open subset of {{mvar|U}} in which {{mvar|T}} vanishes.}}