Almost all positive continuous linear functionals can be extended

TL;DR

This paper demonstrates that in certain ordered topological vector spaces, almost all positive continuous linear functionals on a subspace can be extended to the entire space, highlighting the density of extendable functionals.

Contribution

It proves the density of extendable positive continuous linear functionals in the dual space and shows limitations for general positive operators.

Findings

Positive continuous linear functionals are densely extendable in the dual space.

The result does not extend to arbitrary positive operators, even in finite dimensions.

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-$*$ dense in the topological dual wedge $E_+'$. Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces.