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 0 ≤ v,

$f(v)\geq 0.$

That is to say, a positive linear functional does not necessarily take positive values all the time, but only for positive elements, like the identity function $z\to z$ for complex numbers. 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 C_c(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

$\psi (f)=\int _{X}f(x)d\mu (x)\quad$

for all f in C_c(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.

See also

• positive element