In mathematics, more specifically in functional analysis, a positive linear operator from an preordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} into a preordered vector space ( Y , ≤ ) {\displaystyle (Y,\leq )} 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 ( x ) ≥ 0. {\displaystyle f(x)\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.
Definition A linear function f {\displaystyle f} on a preordered vector space is called positive if it satisfies either of the following equivalent conditions:
x ≥ 0 {\displaystyle x\geq 0} implies f ( x ) ≥ 0. {\displaystyle f(x)\geq 0.}
if x ≤ y {\displaystyle x\leq y} then f ( x ) ≤ f ( y ) . {\displaystyle f(x)\leq f(y).}
The set of all positive linear forms on a vector space with positive cone C , {\displaystyle C,} called the dual cone and denoted by C ∗ , {\displaystyle C^{*},} is a cone equal to the polar of − C . {\displaystyle -C.} The preorder induced by the dual cone on the space of linear functionals on X {\displaystyle X} is called the dual preorder. The order dual of an ordered vector space X {\displaystyle X} is the set, denoted by X + , {\displaystyle X^{+},} defined by X + := C ∗ − C ∗ . {\displaystyle X^{+}:=C^{*}-C^{*}.}
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