Abstract. A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called "order partition spaces," an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact A', and conversely each such CC{X) is an order partition space. {Ce{X) denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of CC{X) for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.A classical theorem of Kakutani (see [10]) states that each order unit space with its order unit topology is lattice isometric to a dense subspace of C(X) containing the constant functions with supremum norm for some compact X and, conversely, each such C(X) is an order unit space. (Here C(X) denotes the space of real-valued continuous functions on X.) In § 1 we generalize the concept of an order unit and the topology which it generates and provide an extension of Kakutani's theorem for this setting. Specifically, we consider a vector lattice having a weak order unit e and a partition of e which, in analogy to the order unit case, defines a locally convex topology. Such a space, called an "order partition space," is shown (Theorem 1) to be lattice isomorphic and homeomorphic to a dense subspace of CC(X) containing the constant functions for some locally compact X. (Here CC(X) denotes C(X) with the topology of compact convergence.) Conversely, CC(X) for X locally compact is an order partition space. The authors obtained a similar result in Theorem 1 of [5] for the restricted case that X is a countable union of open compact sets. In the remainder of §1, sufficient conditions are discussed for the order partition topology to be lattice-theoretic-i.e., independent of the choice of e
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.
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