On Dominated Extension of Linear Operators

An ordered topological vector space has the countable dominated extension property if any linear operator ranging in this space, defined on a subspace of a separable metrizable topological vector space, and dominated there by a continuous sublinear operator admits extension to the entire space with preservation of linearity and domination. Our main result is that the strong $\sigma$-interpolation property is a necessary and sufficient condition for a sequentially complete topological vector space ordered by a closed normal reproducing cone to have the countable dominated extension property. Moreover, this fact can be proved in Zermelo–Fraenkel set theory with the axiom of countable choice.