Inner Product Groups and Riesz Representation Theorem

Abstract

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Keywords

• inner product groups
• Hilbert groups
• Riesz representation theorem
• midconvex subgroups
• normed groups
• torsion-free groups