In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space is a linear functional 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 is a C*-algebra This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.

Sufficient conditions for continuity of all positive linear functionals

There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous. This includes all topological vector lattices that are sequentially complete.

Theorem be an Ordered topological vector space with positive cone is continuous:

1. ).

2. is complete and metrizable

3. is bornological is a semi-complete \mathcal{B}-cone

4. is the inductive limit of ordered Fréchet spaces

Continuous positive extensions

The following theorem is due to H. Bauer and independently, to Namioka.

1. Theorem: be an ordered topological vector space

   Corollary: be an ordered topological vector space

   Corollary: be an ordered vector space if and only if there exists some convex absorbing subset

Examples

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 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.

Positive linear functionals (C*-algebras)

be a C*-algebra (more generally, an operator system

is said to be

1. Theorem.

Cauchy–Schwarz inequality

then one may define a semidefinite sesquilinear form Thus from the Cauchy–Schwarz inequality

Applications to economics

.

See also

References

Bibliography

• Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. .