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In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space {\displaystyle } into a preordered vector space {\displaystyle } is a linear operator f {\displaystyle f} on X {\displaystyle X} into Y {\displaystyle Y} such that for all positive elements x {\displaystyle x} of X , {\displaystyle X,} that is x ≥ 0 , {\displaystyle x\geq 0,} it holds that f ≥ 0. {\displaystyle f\geq 0.} In other words, a positive linear operator maps the positive cone of the 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.