An order topology in ordered topological vector spaces

Abstract

An order topology Ω \Omega that can be defined on any partially-ordered space has as its closed sets those that contain the (o)-limits of all their (o)-convergent nets. In this paper we study the situation in which a topological vector space with a Schauder basis is ordered by the basis cone. In a Fréchet space ( E , τ ) (E,\tau ) , we obtain necessary and sufficient conditions both for τ ⊂ Ω \tau \subset \Omega and for τ = Ω \tau = \Omega . Characterizations of (o)- and Ω \Omega -convergence and of Ω \Omega -closed sets are obtained. The equality of the order topology with the strong topology in certain dual Banach spaces is related to weak sequential completeness through the concept of a shrinking basis.