Inmathematics, more specifically infunctional analysis, apositive linear functional on anordered vector space${\displaystyle (V,\le )}$ is alinear functional${\displaystyle f}$ on${\displaystyle V}$ so that for allpositive elements${\displaystyle v\in V,}$ that is${\displaystyle v\ge 0,}$ it holds that$${\displaystyle f(v)\ge 0.}$$

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 asRiesz–Markov–Kakutani representation theorem.

When${\displaystyle V}$ is acomplex vector space, it is assumed that for all${\displaystyle v\ge 0,}$${\displaystyle f(v)}$ is real. As in the case when${\displaystyle V}$ is aC*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace${\displaystyle W\subseteq V,}$ and the partial order does not extend to all of${\displaystyle V,}$ in which case the positive elements of${\displaystyle V}$ are the positive elements of${\displaystyle W,}$ by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any${\displaystyle x\in V}$ equal to${\displaystyle s^{\ast }s}$ for some${\displaystyle s\in V}$ to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such${\displaystyle x.}$ This property is exploited in theGNS construction to relate positive linear functionals on a C*-algebra toinner products.

There is a comparatively large class ofordered topological vector spaces on which every positive linear form is necessarily continuous.^[1] This includes alltopological vector lattices that aresequentially complete.^[1]

Theorem Let${\displaystyle X}$ be anOrdered topological vector space withpositive cone${\displaystyle C\subseteq X}$ and let${\displaystyle \mathcal{B}\subseteq \mathcal{P}(X)}$ denote the family of all bounded subsets of${\displaystyle X.}$ Then each of the following conditions is sufficient to guarantee that every positive linear functional on${\displaystyle X}$ is continuous:

1. ${\displaystyle C}$ has non-empty topological interior (in${\displaystyle X}$).^[1]
2. ${\displaystyle X}$ iscomplete andmetrizable and${\displaystyle X=C-C.}$^[1]
3. ${\displaystyle X}$ isbornological and${\displaystyle C}$ is asemi-completestrict${\displaystyle \mathcal{B}}$-cone in${\displaystyle X.}$^[1]
4. ${\displaystyle X}$ is theinductive limit of a family${\displaystyle {\left(X_{\alpha }\right)}_{\alpha \in A}}$ of orderedFréchet spaces with respect to a family of positive linear maps where${\displaystyle X_{\alpha }=C_{\alpha }-C_{\alpha }}$ for all${\displaystyle \alpha \in A,}$ where${\displaystyle C_{\alpha }}$ is the positive cone of${\displaystyle X_{\alpha }.}$^[1]

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

Theorem:^[1] Let${\displaystyle X}$ be anordered topological vector space (TVS) with positive cone${\displaystyle C,}$ let${\displaystyle M}$ be a vector subspace of${\displaystyle E,}$ and let${\displaystyle f}$ be a linear form on${\displaystyle M.}$ Then${\displaystyle f}$ has an extension to a continuous positive linear form on${\displaystyle X}$ if and only if there exists some convex neighborhood${\displaystyle U}$ of${\displaystyle 0}$ in${\displaystyle X}$ such that${\displaystyle \mathrm{Re}f}$ is bounded above on${\displaystyle M\cap (U-C).}$

Corollary:^[1] Let${\displaystyle X}$ be anordered topological vector space with positive cone${\displaystyle C,}$ let${\displaystyle M}$ be a vector subspace of${\displaystyle E.}$ If${\displaystyle C\cap M}$ contains an interior point of${\displaystyle C}$ then every continuous positive linear form on${\displaystyle M}$ has an extension to a continuous positive linear form on${\displaystyle X.}$

Corollary:^[1] Let${\displaystyle X}$ be anordered vector space with positive cone${\displaystyle C,}$ let${\displaystyle M}$ be a vector subspace of${\displaystyle E,}$ and let${\displaystyle f}$ be a linear form on${\displaystyle M.}$ Then${\displaystyle f}$ has an extension to a positive linear form on${\displaystyle X}$ if and only if there exists some convexabsorbing subset${\displaystyle W}$ in${\displaystyle X}$ containing the origin of${\displaystyle X}$ such that${\displaystyle \mathrm{Re}f}$ is bounded above on${\displaystyle M\cap (W-C).}$

Proof: It suffices to endow${\displaystyle X}$ with the finest locally convex topology making${\displaystyle W}$ into a neighborhood of${\displaystyle 0\in X.}$

Consider, as an example of${\displaystyle V,}$ the C*-algebra ofcomplexsquare matrices with the positive elements being thepositive-definite matrices. Thetrace function defined on this C*-algebra is a positive functional, as theeigenvalues of any positive-definite matrix are positive, and so its trace is positive.

Consider theRiesz space${\displaystyle {\mathrm{C}}_{\mathrm{c}}(X)}$ of allcontinuous complex-valued functions ofcompactsupport on alocally compactHausdorff space${\displaystyle X.}$ Consider aBorel regular measure${\displaystyle \mu }$ on${\displaystyle X,}$ and a functional${\displaystyle \psi }$ defined by$${\displaystyle \psi (f)={\int }_{X}f(x)d\mu (x)\text{ for all }f\in {\mathrm{C}}_{\mathrm{c}}(X).}$$ 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 theRiesz–Markov–Kakutani representation theorem.

Let${\displaystyle M}$ be a C*-algebra (more generally, anoperator system in a C*-algebra${\displaystyle A}$) with identity${\displaystyle 1.}$ Let${\displaystyle M^{+}}$ denote the set of positive elements in${\displaystyle M.}$

A linear functional${\displaystyle \rho }$ on${\displaystyle M}$ is said to bepositive if${\displaystyle \rho (a)\ge 0,}$ for all${\displaystyle a\in M^{+}.}$

Theorem. A linear functional${\displaystyle \rho }$ on${\displaystyle M}$ is positive if and only if${\displaystyle \rho }$ is bounded and${\displaystyle \Vert \rho \Vert =\rho (1).}$^[2]

If${\displaystyle \rho }$ is a positive linear functional on a C*-algebra${\displaystyle A,}$ then one may define a semidefinitesesquilinear form on${\displaystyle A}$ by${\displaystyle ⟨a,b⟩=\rho (b^{\ast }a).}$ Thus from theCauchy–Schwarz inequality we have$${\displaystyle {\left|\rho (b^{\ast }a)\right|}^2\le \rho (a^{\ast }a)\cdot \rho (b^{\ast }b).}$$

Given a space${\displaystyle C}$, a price system can be viewed as a continuous, positive, linear functional on${\displaystyle C}$.

• Positive element – Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Positive linear operator – Concept in functional analysis

• 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. Vol. 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.

• Functional analysis
• Linear functionals

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