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 ${\displaystyle X}$  be an Ordered topological vector space with positive 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. ${\displaystyle X}$  is complete and metrizable and ${\displaystyle X=C-C.}$ ^[1]
3. ${\displaystyle X}$  is bornological and ${\displaystyle C}$  is a semi-complete strict ${\displaystyle {\mathcal {B}}}$ -cone in ${\displaystyle X.}$ ^[1]
4. ${\displaystyle X}$  is the inductive limit of a family ${\displaystyle \left(X_{\alpha }\right)_{\alpha \in A}}$  of ordered Fréchet 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 an ordered 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 ${\displaystyle \operatorname {Re} f}$  is bounded above on ${\displaystyle M\cap (U-C).}$ 

Corollary:^[1] Let ${\displaystyle X}$  be an ordered 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 an ordered 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 convex absorbing subset ${\displaystyle W}$  in ${\displaystyle X}$  containing the origin of ${\displaystyle X}$  such that ${\displaystyle \operatorname {Re} f}$  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 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 ${\displaystyle \mathrm {C} _{\mathrm {c} }(X)}$  of all continuous complex-valued functions of compact support on a locally compact Hausdorff space ${\displaystyle X.}$  Consider a Borel regular measure ${\displaystyle \mu }$  on ${\displaystyle X,}$  and a functional ${\displaystyle \psi }$  defined by

Let ${\displaystyle M}$  be a C*-algebra (more generally, an operator 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 be positive if ${\displaystyle \rho (a)\geq 0,}$  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 ${\displaystyle \|\rho \|=\rho (1).}$ ^[2]

Cauchy–Schwarz inequality edit

If ${\displaystyle \rho }$  is a positive linear functional on a C*-algebra ${\displaystyle A,}$  then one may define a semidefinite sesquilinear form on ${\displaystyle A}$  by ${\displaystyle \langle a,b\rangle =\rho (b^{\ast }a).}$  Thus from the Cauchy–Schwarz inequality we have

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

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• Positive linear operator – Concept in functional analysis

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