In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space
so that for all positive elements
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.
is a complex vector space, it is assumed that for all
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
This implies that for a C*-algebra, a positive linear functional sends any
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.
be an Ordered topological vector space with positive cone
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
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.
of all continuous complex-valued functions of compact support on a locally compact Hausdorff space
Consider a Borel regular measure
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.
be a C*-algebra (more generally, an operator system in a C*-algebra
denote the set of positive elements in
is a positive linear functional on a C*-algebra
then one may define a semidefinite sesquilinear form on
, a price system can be viewed as a continuous, positive, linear functional on