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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'} .
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 .
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 +.
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 ).
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 .
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 ...
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