Extending a positive linear functional in finite dimensions Let $V$ be a vector subspace of $R^N$, and $l:V \to R$ a linear mapping such that $l(V\bigcap R_{+}^N)\subseteq R_{+}$ (i.e., $l$ is positive).
I have heard that there exists a separating hyperplane sort of argument that allows us to show that $l$ extends to a positive linear functional on $R^N$. I have my own proof, but it is complicated and uses the Bauer-Namioka condition for extension of positive linear functionals on ordered vector spaces of any dimension.
Question: What is this simple separating hyperplane argument that shows that $l$ extends to a positive linear functional on $R^N$? (I believe it should be quite straightforward, but I was not able to construct the right problem to invoke a separation argument...) A reference would be okay as well. Thanks.
 A: (Edit: reading the comments left by Kevin above, Riesz extension requires a bit more than he assumed. But I think that the separating hyperplane argument would boil down to essentially the same trick as outlined below.)
I think you can use the following modified proof of the Riesz extension theorem.
Marcel Riesz Extension Theorem Let $W$ be a vector space (finite dimensional in the following proof; the actual theorem has no such limitations), and $V$ a subspace. Let $F$ be a convex cone in $W$ with the property that for every $w\in W$ there exists $v_+, v_- \in V$ such that $v_+ - w\in F$ and $w - v_- \in F$. Then for any $\phi$ linear on $V$ and positive on $V\cap F$, there exists an extension $\psi$ on $W$ that is positive on $F$. 
Proof (sketch) Define $\psi_+(w) = \inf_{v-w\in F, v\in V}\phi(v)$ and $\psi_-(w) = \sup_{w-v\in F, v\in V}\phi(v)$. Check that $\psi_+$ is convex, and $\psi_-(w)$ is concave, and $\psi_+(w) \geq \psi_-(w)$. An application of the separating hyperplane theorem implies that there exists a linear map $\psi$ with $\psi_+ \geq \psi \geq \psi_-$. Since $\psi_+ = \psi_- |_V$, you have that $\psi$ is an extension. By definition $\psi_-|_F > 0$, and so you have the conclusion of the theorem. 
(See also this post on Terry Tao's blog for some related concepts.)
A: Perhaps this will help (Exercise 12 seems similar).
