# Positive linear functional

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.^{[clarification needed]} 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.

## Sufficient conditions for continuity of all positive linear functionals[edit]

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:

- has non-empty topological interior (in ).
^{[1]} - is complete and metrizable and .
^{[1]} - is bornological and is a semi-complete strict -cone in .
^{[1]} - is the inductive limit of a family of ordered Fréchet spaces with respect to a family of positive linear maps where for all , where is the positive cone of .
^{[1]}

## Continuous positive extensions[edit]

The following theorem is due to H. Bauer and independently, to Namioka.^{[1]}

**Theorem**:^{[1]}Let be an ordered topological vector space (TVS) with positive cone , let be a vector subspace of , and let be a linear form on . Then has an extension to a continuous positive linear form on if and only if there exists some convex neighborhood of such that is bounded above on .

**Corollary**:^{[1]}Let be an ordered topological vector space with positive cone , let be a vector subspace of . If contains an interior point of then every continuous positive linear form on has an extension to a continuous positive linear form on .

**Corollary**:^{[1]}Let be an ordered vector space with positive cone , let be a vector subspace of , and let be a linear form on . Then has an extension to a positive linear form on if and only if there exists some convex absorbing subset in containing such that is bounded above on .

Proof: It suffices to endow with the finest locally convex topology making into a neighborhood of .

## Examples[edit]

- 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

- for all in . 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)[edit]

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 .

**Theorem.**A linear functional on is positive if and only if is bounded and .^{[2]}

### Cauchy–Schwarz inequality[edit]

If ρ 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

## See also[edit]

## References[edit]

## Bibliography[edit]

- Kadison, Richard,
*Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory*, American Mathematical Society. ISBN 978-0821808191. - Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.