Abstract
In this article, a completion theorem for cone metric spaces and a completion theorem for cone normed spaces are proved. The completion spaces are constructed by means of an equivalence relation defined via an ordered cone norm on the Banach space
E
whose cone is strongly minihedral and ordered closed. This order norm has to satisfy the generalized absolute value property. In particular, if
E
is a Dedekind complete Banach lattice, then, together with its absolute value norm, satisfy the desired properties.