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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.
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 ).
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 +.
[2] One approach to measure theory is to start with a Radon measure, defined as a positive linear functional on C c (X). This is the way adopted by Bourbaki; it does of course assume that X starts life as a topological space, rather than simply as a set. For locally compact spaces an integration theory is then recovered.
A linear function on a preordered vector space is called positive if it satisfies either of the following equivalent conditions: . implies (); if then () (). [1]; The set of all positive linear forms on a vector space with positive cone , called the dual cone and denoted by , is a cone equal to the polar of .
In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices).Phrased differently, the duality is the isomorphism between completely positive maps (channels) from A to C n×n, where A is a C*-algebra and C n×n denotes the n×n complex entries, and positive linear functionals on the tensor product
The states of the quantum mechanical system are defined to be the states of the C*-algebra (in other words the normalized positive linear functionals ). The value ω ( A ) {\displaystyle \omega (A)} of a state ω {\displaystyle \omega } on an element A {\displaystyle A} is the expectation value of the observable A {\displaystyle A} if the ...