Representation of a class of locally convex (M)-lattices

We prove a representation theorem for Hausdorff locally convex (M)-lattices which are Dedekind σ-complete, and whose topologies are order σ-continuous and monotonically complete. These turn out to be the weighted spaces c0(T, H), defined in the paper for T ≠ ∅ and H ⊂ ℝT+. We also characterize the dual of c0(T, H), as the space l1 (T, H) defined in the last section. The known representation (on c0(T)) of Banach (M)-lattices with order continuous norm follows as a particular case. We obtain these results by first proving a new general isomorphism theorem, which seems to be of independent interest. Our notion of “monotonic topological completeness” is weaker than the usual completeness and seems to be very convenient in the framework of topological ordered vector spaces.