An Order-Theoretic Description of Marinescu Spaces

Abstract

Abstract.It is shown that any inductive limit E in the category of convergence spaces of real locally convex topological vector spaces (i.e., any Marinescu space) can be embedded in a partially ordered vector space so that convergence in E can be characterized as an order-theoretic convergence. The ordertheoretic convergence in question is a modification of classical order convergence.Introduction. DeMarr [1] has shown that every real Hausdorff locally convex space V can be embedded in a partially ordered vector space so that the convergence in V can be realized as unbounded order convergence. In the theorem of §2 we provide an analogous order-theoretic description of real Marinescu spaces. Marinescu spaces include locally convex spaces (not necessarily Hausdorff) as well as important examples of nontopological convergence vector spaces ; pertinent examples can be found in [2] and [3]. It is of interest to note that the theorem provides an order-theoretic characterization of almost-everywhere convergence of nets of measurable functions on [0, 1] (see §1). This nontopological convergence, when restricted to sequences, has previously been described order-theoretically (e.g., see [1] and [4]), but these characterizations do not extend to arbitrary nets.