A characterization of the Dedekind completion of a totally ordered group of infinite rank *

In non-Archimedean functional analysis the Dedekind completion of a linearly ordered group of infinite rank is an important object, being the natural home for the norms of vectors as well as of linear operators. However the standard construction by cuts does not give the much needed actual description of the elements obtained. In this paper we consider a class of Hahn products, called Λα (α an ordinal), whose rank is the order-type of α. We give an operational representation of every element of the Dedekind completion of such a group in terms of the supremum and infimum of its convex subgroups.