Dual spaces of a vector lattice and its cut-completion

Abstract

Introduction. Let L be an Archimedean vector lattice. The space Ln of all normal integrals (order continuous linear functionals) on L was first studied by H. Nakano [6], and has been the object of recent investigation [1]-[5]. The central theorem in this paper shows that if L is the cut-completion of L, then Ln and (L)n are isomorphic vector lattices, a known result when L= QX), the continuous functions on a compact Hausdorff space (cf. [3, ?10]). This serves as a convenient tool for examining the role that completeness of L plays in determining properties of Ln. It will be shown that a number of results previously obtained under the assumption that L is complete can be extended to Archimedean vector lattices.