|
|
| Line 1: |
Line 1: |
| {{Unreferenced|date=December 2009}}
| | Howdy! I am Dalton. Acting should be a thing that totally addicted to. My carry is now in Vermont and I don't program on changing it. I am a cashier. I'm not sensible at webdesign but you can want to check my website: http://prometeu.net<br><br>Here is my website: clash of clans hack no survey ([http://prometeu.net what is it worth]) |
| In [[mathematics]], especially 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 element (ordered group)|positive element]]s ''v'' of ''V'', that is ''v''≥0, it holds that
| |
| | |
| :<math>f(v)\geq 0.</math>
| |
| | |
| 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]].
| |
| | |
| ==Examples==
| |
| * Consider, as an example of V, the C*-algebra of [[complex number|complex]] [[square matrix|square matrices]] with the positive elements being the [[positive-definite matrix|positive-definite matrices]]. The [[trace of a matrix|trace]] function defined on this C*-algebra is a positive functional, as the [[eigenvalue]]s of any positive-definite matrix are positive, and so its trace is positive.
| |
| | |
| * Consider the [[Riesz space]] C<sub>c</sub>(''X'') of all [[continuous function (topology)|continuous]] complex-valued functions of [[compact space|compact]] [[support (mathematics)|support]] on a [[locally compact]] [[Hausdorff space]] ''X''. Consider a [[Borel regular measure]] μ on ''X'', and a functional ψ defined by
| |
| ::<math> \psi(f) = \int_X f(x) d \mu(x) \quad </math> | |
| :for all ''f'' in C<sub>c</sub>(''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]].
| |
| | |
| ==See also==
| |
| *[[positive element (ordered group)|positive element]]
| |
| | |
| {{Functional Analysis}}
| |
| | |
| {{DEFAULTSORT:Positive Linear Functional}}
| |
| [[Category:Functional analysis]]
| |
Howdy! I am Dalton. Acting should be a thing that totally addicted to. My carry is now in Vermont and I don't program on changing it. I am a cashier. I'm not sensible at webdesign but you can want to check my website: http://prometeu.net
Here is my website: clash of clans hack no survey (what is it worth)