Editing Distribution (mathematics) (section)

===Extensions and restrictions to an open subset===

Let <math>V \subseteq U</math> be open subsets of <math>\R^n.</math> 
Every function <math>f \in \mathcal{D}(V)</math> can be {{em|extended by zero}} from its domain {{mvar|V}} to a function on {{mvar|U}} by setting it equal to <math>0</math> on the [[Complement (set theory)|complement]] <math>U \setminus V.</math> This extension is a smooth compactly supported function called the {{em|trivial extension of <math>f</math> to <math>U</math>}} and it will be denoted by <math>E_{VU} (f).</math>
This assignment <math>f \mapsto E_{VU} (f)</math> defines the {{em|trivial extension}} operator 
<math>E_{VU} : \mathcal{D}(V) \to \mathcal{D}(U),</math> 
which is a continuous injective linear map. It is used to canonically identify <math>\mathcal{D}(V)</math> as a [[vector subspace]] of <math>\mathcal{D}(U)</math> (although {{em|not}} as a [[topological subspace]]). 
Its transpose ([[#Transpose of a linear operator|explained here]]) 
<math display=block>\rho_{VU} := {}^{t}E_{VU} : \mathcal{D}'(U) \to \mathcal{D}'(V),</math> 
is called the '''{{em|{{visible anchor|restriction map|text=restriction to <math>V</math> of distributions in <math>U</math>}}}}'''{{sfn|Trèves|2006|pp=245-247}} and as the name suggests, the image <math>\rho_{VU}(T)</math> of a distribution <math>T \in \mathcal{D}'(U)</math> under this map is a distribution on <math>V</math> called the '''restriction of <math>T</math> to <math>V.</math>''' The [[#Transpose of a linear operator|defining condition]] of the restriction <math>\rho_{VU}(T)</math> is:
<math display=block>\langle \rho_{VU} T, \phi \rangle = \langle T, E_{VU} \phi \rangle \quad \text{ for all } \phi \in \mathcal{D}(V).</math>
If <math>V \neq U</math> then the (continuous injective linear) trivial extension map <math>E_{VU} : \mathcal{D}(V) \to \mathcal{D}(U)</math> is {{em|not}} a topological embedding (in other words, if this linear injection was used to identify <math>\mathcal{D}(V)</math> as a subset of <math>\mathcal{D}(U)</math> then <math>\mathcal{D}(V)</math>'s topology would [[Comparison of topologies|strictly finer]] than the [[subspace topology]] that <math>\mathcal{D}(U)</math> induces on it; importantly, it would {{em|not}} be a [[topological subspace]] since that requires equality of topologies) and its range is also {{em|not}} dense in its [[codomain]] <math>\mathcal{D}(U).</math>{{sfn|Trèves|2006|pp=245-247}} Consequently if <math>V \neq U</math> then [[#restriction map|the restriction mapping]] is neither injective nor surjective.{{sfn|Trèves|2006|pp=245-247}} A distribution <math>S \in \mathcal{D}'(V)</math> is said to be '''{{em|extendible to {{mvar|U}}}}''' if it belongs to the range of the transpose of <math>E_{VU}</math> and it is called '''{{em|extendible}}''' if it is extendable to <math>\R^n.</math>{{sfn|Trèves|2006|pp=245-247}}

Unless <math>U = V,</math> the restriction to {{mvar|V}} is neither [[injective]] nor [[surjective]]. Lack of surjectivity follows since distributions can blow up towards the boundary of {{mvar|V}}. For instance, if <math>U = \R</math> and <math>V = (0, 2),</math> then the distribution
<math display=block>T(x) = \sum_{n=1}^\infty n \, \delta\left(x-\frac{1}{n}\right)</math>
is in <math>\mathcal{D}'(V)</math> but admits no extension to <math>\mathcal{D}'(U).</math>