Editing Distribution (mathematics) (section)

===Canonical LF topology===
{{Main|Spaces of test functions and distributions}}
{{See also|LF-space|Topology of uniform convergence}}

Recall that <math>C_c^k(U)</math> denotes all functions in <math>C^k(U)</math> that have compact [[#support of a function|support]] in <math>U,</math> where note that <math>C_c^k(U)</math> is the union of all <math>C^k(K)</math> as <math>K</math> ranges over all compact subsets of <math>U.</math> Moreover, for each <math>k,\, C_c^k(U)</math> is a dense subset of <math>C^k(U).</math> The special case when <math>k = \infty</math> gives us the space of test functions.

{{block indent|em=1.5|text=<math>C_c^\infty(U)</math> is called the {{em|'''space of test functions''' on <math>U</math>}} and it may also be denoted by <math>\mathcal{D}(U).</math> Unless indicated otherwise, it is endowed with a topology called '''{{em|the canonical LF topology}}''', whose definition is given in the article: [[Spaces of test functions and distributions]].}}

The canonical LF-topology is {{em|not}} metrizable and importantly, it is [[Comparison of topologies|{{em|'''strictly''' finer}}]] than the [[subspace topology]] that <math>C^\infty(U)</math> induces on <math>C_c^\infty(U).</math> However, the canonical LF-topology does make <math>C_c^\infty(U)</math> into a [[Complete topological vector space|complete]] [[Reflexive space|reflexive]] [[Nuclear space|nuclear]]{{sfn|Trèves|2006|pp=526-534}} [[Montel space|Montel]]{{sfn|Trèves|2006|p=357}} [[Bornological space|bornological]] [[Barrelled space|barrelled]] [[Mackey space]]; the same is true of its [[strong dual space]] (that is, the space of all distributions with its usual topology). The canonical [[LF-space|LF-topology]] can be defined in various ways.