In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional on so that for all positive elements that is 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 is a complex vector space, it is assumed that for all is real. As in the case when is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace and the partial order does not extend to all of in which case the positive elements of are the positive elements of by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any equal to for some to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.
There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous.[1] This includes all topological vector lattices that are sequentially complete.[1]
Theorem Let be an Ordered topological vector space with positive cone and let denote the family of all bounded subsets of Then each of the following conditions is sufficient to guarantee that every positive linear functional on is continuous:
The following theorem is due to H. Bauer and independently, to Namioka.[1]
Proof: It suffices to endow with the finest locally convex topology making into a neighborhood of
Consider, as an example of 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 of all continuous complex-valued functions of compact support on a locally compact Hausdorff space Consider a Borel regular measure on and a functional defined by
Let be a C*-algebra (more generally, an operator system in a C*-algebra ) with identity Let denote the set of positive elements in
A linear functional on is said to be positive if for all
If is a positive linear functional on a C*-algebra then one may define a semidefinite sesquilinear form on by Thus from the Cauchy–Schwarz inequality we have
Given a space , a price system can be viewed as a continuous, positive, linear functional on .