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In [[mathematics]], more specifically in [[functional analysis]], a '''positive linear functional''' on an [[ordered vector space]] <math>(V, \leq)</math> is a [[linear functional]] <math>f</math> on <math>V</math> so that for all [[positive element (ordered group)|positive element]]s <math>v \in V,</math> that is <math>v \geq 0,</math> it holds that
<math display=block>f(v) \geq 0.</math>

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 <math>V</math> is a [[Complex numbers|complex]] vector space, it is assumed that for all <math>v\ge0,</math> <math>f(v)</math> is real. As in the case when <math>V</math> is a [[C*-algebra]] with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace <math>W\subseteq V,</math> and the partial order does not extend to all of <math>V,</math> in which case the positive elements of <math>V</math> are the positive elements of <math>W,</math> by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any <math>x \in V</math> equal to <math>s^{\ast}s</math> for some <math>s \in V</math> to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such <math>x.</math> This property is exploited in the [[GNS construction]] to relate positive linear functionals on a C*-algebra to [[inner product]]s.

== Sufficient conditions for continuity of all positive linear functionals ==

There is a comparatively large class of [[ordered topological vector space]]s on which every positive linear form is necessarily continuous.{{sfn|Schaefer|Wolff|1999|pp=225-229}}
This includes all [[topological vector lattice]]s that are [[Sequentially complete topological vector space|sequentially complete]].{{sfn|Schaefer|Wolff|1999|pp=225-229}}

'''Theorem''' Let <math>X</math> be an [[Ordered topological vector space]] with [[positive cone of an ordered vector space|positive cone]] <math>C \subseteq X</math> and let <math>\mathcal{B} \subseteq \mathcal{P}(X)</math> denote the family of all bounded subsets of <math>X.</math>
Then each of the following conditions is sufficient to guarantee that every positive linear functional on <math>X</math> is continuous:
# <math>C</math> has non-empty topological interior (in <math>X</math>).{{sfn|Schaefer|Wolff|1999|pp=225-229}}
# <math>X</math> is [[Complete space|complete]] and [[metrizable]] and <math>X = C - C.</math>{{sfn|Schaefer|Wolff|1999|pp=225-229}}
# <math>X</math> is [[Bornological space|bornological]] and <math>C</math> is a [[semi-complete]] [[Normal cone (functional analysis)|strict <math>\mathcal{B}</math>-cone]] in <math>X.</math>{{sfn|Schaefer|Wolff|1999|pp=225-229}}
# <math>X</math> is the [[inductive limit]] of a family <math>\left(X_{\alpha} \right)_{\alpha \in A}</math> of ordered [[Fréchet space]]s with respect to a family of positive linear maps where <math>X_{\alpha} = C_{\alpha} - C_{\alpha}</math> for all <math>\alpha \in A,</math> where <math>C_{\alpha}</math> is the positive cone of <math>X_{\alpha}.</math>{{sfn|Schaefer|Wolff|1999|pp=225-229}}

== Continuous positive extensions ==

The following theorem is due to H. Bauer and independently, to Namioka.{{sfn|Schaefer|Wolff|1999|pp=225-229}}

:'''Theorem''':{{sfn|Schaefer|Wolff|1999|pp=225-229}} Let <math>X</math> be an [[ordered topological vector space]] (TVS) with positive cone <math>C,</math> let <math>M</math> be a vector subspace of <math>E,</math> and let <math>f</math> be a linear form on <math>M.</math> Then <math>f</math> has an extension to a continuous positive linear form on <math>X</math> if and only if there exists some convex neighborhood <math>U</math> of <math>0</math> in <math>X</math> such that <math>\operatorname{Re} f</math> is bounded above on <math>M \cap (U - C).</math>

:'''Corollary''':{{sfn|Schaefer|Wolff|1999|pp=225-229}} Let <math>X</math> be an [[ordered topological vector space]] with positive cone <math>C,</math> let <math>M</math> be a vector subspace of <math>E.</math> If <math>C \cap M</math> contains an interior point of <math>C</math> then every continuous positive linear form on <math>M</math> has an extension to a continuous positive linear form on <math>X.</math>

:'''Corollary''':{{sfn|Schaefer|Wolff|1999|pp=225-229}} Let <math>X</math> be an [[ordered vector space]] with positive cone <math>C,</math> let <math>M</math> be a vector subspace of <math>E,</math> and let <math>f</math> be a linear form on <math>M.</math> Then <math>f</math> has an extension to a positive linear form on <math>X</math> if and only if there exists some convex [[Absorbing set|absorbing subset]] <math>W</math> in <math>X</math> containing the origin of <math>X</math> such that <math>\operatorname{Re} f</math> is bounded above on <math>M \cap (W - C).</math>

Proof: It suffices to endow <math>X</math> with the finest locally convex topology making <math>W</math> into a neighborhood of <math>0 \in X.</math>

== Examples ==

Consider, as an example of <math>V,</math> the C*-algebra of [[Complex number|complex]] [[Square matrix|square matrices]] with the positive elements being the [[Positive-definite matrix|positive-definite matrices]]. The [[Trace of a matrix|trace]] function defined on this C*-algebra is a positive functional, as the [[eigenvalue]]s of any positive-definite matrix are positive, and so its trace is positive.

Consider the [[Riesz space]] <math>\mathrm{C}_{\mathrm{c}}(X)</math> of all [[Continuous function (topology)|continuous]] complex-valued functions of [[Compact space|compact]] [[Support (mathematics)|support]] on a [[locally compact]] [[Hausdorff space]] <math>X.</math> Consider a [[Borel regular measure]] <math>\mu</math> on <math>X,</math> and a functional <math>\psi</math> defined by <math display=block>\psi(f) = \int_X f(x) d \mu(x) \quad \text{ for all } f \in \mathrm{C}_{\mathrm{c}}(X).</math> 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) ==

Let <math>M</math> be a C*-algebra (more generally, an [[operator system]] in a C*-algebra <math>A</math>) with identity <math>1.</math> Let <math>M^+</math> denote the set of positive elements in <math>M.</math>

A linear functional <math>\rho</math> on <math>M</math> is said to be {{em|positive}} if <math>\rho(a) \geq 0,</math> for all <math>a \in M^+.</math>
:'''Theorem.''' A linear functional <math>\rho</math> on <math>M</math> is positive if and only if <math>\rho</math> is bounded and <math>\|\rho\| = \rho(1).</math><ref name=Murphy>{{cite book|last=Murphy|first=Gerard |title=C*-Algebras and Operator Theory|publisher=Academic Press, Inc.|isbn=978-0125113601|edition=1st|chapter=3.3.4|pages=89}}</ref>

=== Cauchy–Schwarz inequality ===

If <math>\rho</math> is a positive linear functional on a C*-algebra <math>A,</math> then one may define a semidefinite [[sesquilinear form]] on <math>A</math> by <math>\langle a,b\rangle = \rho(b^{\ast}a).</math> Thus from the [[Cauchy–Schwarz inequality]] we have
<math display=block>\left|\rho(b^{\ast}a)\right|^2 \leq \rho(a^{\ast}a) \cdot \rho(b^{\ast}b).</math>

== Applications to economics ==
Given a space <math>C</math>, a price system can be viewed as a continuous, positive, linear functional on <math>C</math>.

== See also ==

* {{annotated link|Positive element (ordered group)|Positive element}}
* {{annotated link|Positive linear operator}}

== References ==

{{reflist|group=note}}
{{reflist}}

==Bibliography==

* [[Kadison, Richard]], ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. {{ISBN|978-0821808191}}.
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} 
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} 
* {{Trèves François Topological vector spaces, distributions and kernels}} 

{{Functional analysis}}
{{Ordered topological vector spaces}}

{{DEFAULTSORT:Positive Linear Functional}}
[[Category:Functional analysis]]
[[Category:Linear functionals]]

Positive linear functional - Revision history

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