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  2. Positive linear functional - Wikipedia

    en.wikipedia.org/wiki/Positive_linear_functional

    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.

  3. Completely positive map - Wikipedia

    en.wikipedia.org/wiki/Completely_positive_map

    Download as PDF; Printable version; ... In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely ...

  4. Gelfand–Naimark–Segal construction - Wikipedia

    en.wikipedia.org/wiki/Gelfand–Naimark–Segal...

    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'} .

  5. State (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/State_(functional_analysis)

    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 ).

  6. Category:Linear functionals - Wikipedia

    en.wikipedia.org/wiki/Category:Linear_functionals

    Download as PDF; Printable version; ... Pages in category "Linear functionals" ... Positive linear functional; R.

  7. Normed vector space - Wikipedia

    en.wikipedia.org/wiki/Normed_vector_space

    Download as PDF; Printable version; ... Positive definiteness: ... An important theorem about continuous linear functionals on normed vector spaces is the Hahn ...

  8. Radon measure - Wikipedia

    en.wikipedia.org/wiki/Radon_measure

    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.

  9. Order dual (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Order_dual_(functional...

    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 +.