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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'} .
Finally, is positive if and only if the measure is positive. One can deduce this statement about linear functionals from the statement about positive linear functionals by first showing that a bounded linear functional can be written as a finite linear combination of positive ones.
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 +.
In linear algebra, it is synonymous with a linear form, which is a linear mapping from a vector space into its field of scalars (that is, it is an element of the dual space) [1] In functional analysis and related fields, it refers to a mapping from a space X {\displaystyle X} into the field of real or complex numbers .
Conversely, by the Riesz–Markov–Kakutani representation theorem, each positive linear form on K (X) arises as integration with respect to a unique regular Borel measure. A real-valued Radon measure is defined to be any continuous linear form on K ( X ) ; they are precisely the differences of two Radon measures.
These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathematical analysis , the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product , norm , or topology ) and the ...