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.
Highlights
Introduction and PreliminariesCitation: Pourmoslemi, A.; Nazari, T.; Salimi, M
In 1936, Birkhoff and Kakutani independently proved a significant theorem: A Hausdorff group K is homeomorphic with a metric space, if and only if K satisfies the first countability axiom
They showed that this group has a right invariant metric
Summary
The normed group (K, k.k) is called complete if any Cauchy sequence in K converges to an element of K; i.e., it has a limit in group K. k v k = k v −1 k (Symmetry); kvsk ≤ kvk + ksk ( Triangle inequality); kvk ≥ 0 and kvk = 0 i f f v = e ( Positivity). Note that k.k is abelian because kr i r j k = kr i + j k = kr j +i k = kr j r i k, kr i s j k = k s i + j k = 2 = ksjrik, 0, i f i = j k si s j k = kri − j k = This shows that there is an abelian norm on a non-abelian group. For a divisible element v in a N-homogeneous normed group K, let sn = v; m m ksk = n1 kvk and as sm = v n , we have mn kvk = kv n k.
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