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In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space (''V'', ≤) is a linear functional ''f'' on ''V'' so that for all positive elements ''v'' of ''V'', that is ''v''≥0, it holds that : In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem. When ''V'' is a complex vector space, it is assumed that for all ''v''≥0 , ''f''(''v'') is real. As in the case when ''V'' is a C *-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace ''W'' of ''V'', and the partial order does not extend to all of ''V'', in which case the positive elements of ''V'' are the positive elements of ''W'', by abuse of notation. This implies that for a C *-algebra, a positive linear functional sends any ''x'' in ''V'' equal to ''s *s'' for some ''s'' in ''V'' to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such ''x''. This property is exploited in the GNS construction to relate positive linear functionals on a C *-algebra to inner products. ==Examples== * Consider, as an example of V, the C *-algebra of complex square matrices with the positive elements being the positive-definite matrices. The trace function defined on this C *-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive. * Consider the Riesz space Cc(''X'') of all continuous complex-valued functions of compact support on a locally compact Hausdorff space ''X''. Consider a Borel regular measure μ on ''X'', and a functional ψ defined by :: :for all ''f'' in Cc(''X''). Then, this functional is positive (the integral of any positive function is a positive number). Moreover, any positive functional on this space has this form, as follows from the Riesz–Markov–Kakutani representation theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「positive linear functional」の詳細全文を読む スポンサード リンク
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