Normed versus topological groups: Dichotomy and duality

12 N. H. Bingham and A. J. OstaszewskiFor further information on norms with the Heine-Borel property (for which compactsets are precisely those sets which are closed in the right norm topology and normbounded)see [?]).The significance of the following simple corollary is wide-ranging. It explicitly demonstratesthat small either-sided translations λ x , ρ y do not much alter the norm. Its maineffect is on the analysis of subadditive functions.Corollary 2.6. With ‖x‖ := d X (x, e), where d X is a right-invariant metric on X,|(‖x‖ − ‖y‖)| ≤ ‖xy‖ ≤ ‖x‖ + ‖y‖.Proof. By Proposition 2.2, the triangle inequality and symmetry holds for norms, so‖y‖ = ‖x −1 xy‖ ≤ ‖x −1 ‖ + ‖xy‖ = ‖x‖ + ‖xy‖.We now generalize (rv-limit), by letting T, X be subgroups of a normed group G withX invariant under T.Definition. We say that a function h : X → H is slowly varying on X over T if∂ X h(t) = e H , that is, for each t in Th(tx)h(x) −1 → e H , as ‖x‖ → ∞ for x ∈ X.We omit mention of X and T when context permits. In practice G will be an internaldirect product of two normal subgroups G = T X. (For a topological view on the internaldirect product, see [Na, Ch. 2.7] ; for an algebraic view see [vdW, Ch. 6, Sect. 47], [J] Ch. 9and 10, or [Ga] Section 9.1.) We may verify the property of h just defined by comparisonwith a slowly varying function.Theorem 2.7 (Comparison criterion). h : X → H is slowly varying iff for some slowlyvarying function g : X → H and some µ ∈ H,lim ‖x‖→∞ h(x)g(x) −1 = µ.Proof. If this holds for some slowly varying g and some µ,h(tx)h(x) −1 = h(tx)g(tx) −1 g(tx)g(x) −1 g(x)h(x) −1 → µe H µ −1 = e H ,so h is slowly varying; the converse is trivial.Theorem 2.8. For d X a right-invariant metric on a group X, the norm ‖x‖ := d X (x, e),as a function from X to the multiplicative positive reals R ∗ +, is slowly varying in themultiplicative sense, i.e., for any t ∈ X,Hence alsolim ‖x‖→∞‖tx‖‖x‖ = 1.lim ‖x‖→∞‖gxg −1 ‖‖x‖= 1.