Almost all positive continuous linear functionals can be extended

Abstract

Let $F$ be an ordered topological vector space (over $\mathbb{R}$) whose positive cone $F_{+}$ is weakly closed, and let $E\subseteq F$ be a subspace. We prove that the set of positive continuous linear functionals on $E$ that can be extended (positively and continuously) to $F$ is weak-$\ast$ dense in the topological dual wedge $E_{+}^{\prime }$. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.

4 pages; changes for v2: improved structure, added references, removed second proof, added corollary