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The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When V {\displaystyle V} is a complex vector space, it is assumed that for all v ≥ 0 , {\displaystyle v\geq 0,} f ( v ) {\displaystyle f(v)} is real.
There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or ...
Any positive linear functionals on dominated by is of the form = (), for some positive operator in () ′ with in the operator order. This is a version of the Radon–Nikodym theorem . For such g {\displaystyle g} , one can write f {\displaystyle f} as a sum of positive linear functionals: f = g + g ′ {\displaystyle f=g+g'} .
A proof can be sketched as follows: Let be the weak*-compact set of positive linear functionals on with norm ≤ 1, and () be the continuous functions on . A {\displaystyle A} can be viewed as a closed linear subspace of C ( Ω ) {\displaystyle C(\Omega )} (this is Kadison 's function representation ).
Positive linear functionals on () correspond to (positive) regular Borel measures on , by a different form of the Riesz representation theorem. ( Rudin 1966 , Chapter 2) If X {\displaystyle X} is infinite, then C ( X ) {\displaystyle {\mathcal {C}}(X)} is not reflexive , nor is it weakly complete .
A weight ω on a von Neumann algebra is a linear map from the set of positive elements (those of the form a*a) to [0,∞]. A positive linear functional is a weight with ω(1) finite (or rather the extension of ω to the whole algebra by linearity). A state is a weight with ω(1) = 1. A trace is a weight with ω(aa*) = ω(a*a) for all a.
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all , implies () [1] The order dual of is denoted by +.
Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous functions with compact support (some authors use this as the definition of a Radon measure). This produces a good theory with no pathological problems ...