Positive linear functional
- Positive linear functional
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 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 representation theorem.
Examples
* Consider the C*-algebra of complex square matrices. Then, the positive elements are 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 representation theorem.
ee also
*positive element
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