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If contains an interior point of then every continuous positive linear form on has an extension to a continuous positive linear form on . Corollary : [ 1 ] Let X {\displaystyle X} be an ordered vector space with positive cone C , {\displaystyle C,} let M {\displaystyle M} be a vector subspace of E , {\displaystyle E,} and let f {\displaystyle f ...
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'} .
[2] [3] In functional analysis, the term linear functional is a synonym of linear form; [3] [4] [5] that is, it is a scalar-valued linear map. Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space . [citation needed]
By Gelfand representation, every commutative C*-algebra A is of the form C 0 (X) for some locally compact Hausdorff X. In this case, S(A) consists of positive Radon measures on X, and the pure states are the evaluation functionals on X. More generally, the GNS construction shows that every state is, after choosing a suitable representation, a ...
The statement of the theorem for positive linear functionals on C c (X), the space of compactly supported complex-valued continuous functions, is as follows: Theorem Let X be a locally compact Hausdorff space and ψ {\displaystyle \psi } a positive linear functional on C c ( X ) .
The positive elements of the order dual form a cone that induces an ordering on + called the canonical ordering. If X {\displaystyle X} is an ordered vector space whose positive cone C {\displaystyle C} is generating (that is, X = C − C {\displaystyle X=C-C} ) then the order dual with the canonical ordering is an ordered vector space. [ 1 ]
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
The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. [1] [2] The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy.