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{{Short description|Concept in functional analysis}}
{{Multiple issues|{{refimprove|date=June 2020}}{{lead rewrite|date=June 2020|reason=The lead should be a summary of the body of the article.}}}}
In [[mathematics]], more specifically in [[functional analysis]], a '''positive linear operator''' from an [[Ordered vector space|preordered vector space]] <math>(X, \leq)</math> into a preordered vector space <math>(Y, \leq)</math> is a [[linear operator]] <math>f</math> on <math>X</math> into <math>Y</math> such that for all [[Positive element (ordered group)|positive element]]s <math>x</math> of <math>X,</math> that is <math>x \geq 0,</math> it holds that <math>f(x) \geq 0.</math>
In other words, a positive linear operator maps the positive cone of the [[Domain of a function|___domain]] into the positive cone of the [[codomain]].
Every [[positive linear functional]] is a type of positive linear operator.
The significance of positive linear operators lies in results such as [[Riesz–Markov–Kakutani representation theorem]].
== Definition ==
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If <math>M</math> is a vector subspace of <math>\mathcal{L}(X; Y)</math> and if <math>H \cap M</math> is a proper cone then this proper cone defines a '''{{visible anchor|canonical partial order}}''' on <math>M</math> making <math>M</math> into a partially ordered vector space.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}
If <math>(X, \leq)</math> and <math>(Y, \leq)</math> are [[ordered topological vector space]]s and if <math>\mathcal{G}</math> is a family of bounded subsets of <math>X</math> whose union covers <math>X</math> then the [[positive cone of an ordered vector space|positive cone]] <math>\mathcal{H}</math> in <math>L(X; Y)</math>, which is the space of all continuous linear maps from <math>X</math> into <math>Y,</math> is closed in <math>L(X; Y)</math> when <math>L(X; Y)</math> is endowed with the [[Topology of uniform convergence|<math>\mathcal{G}</math>-topology]].{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}
For <math>\mathcal{H}</math> to be a proper cone in <math>L(X; Y)</math> it is sufficient that the positive cone of <math>X</math> be total in <math>X</math> (that is, the span of the positive cone of <math>X</math> be dense in <math>X</math>).
If <math>Y</math> is a locally convex space of dimension greater than 0 then this condition is also necessary.{{sfn | Schaefer | Wolff | 1999 | pp=225–229}}
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