Normed versus topological groups: Dichotomy and duality

Normed groups 55equipped with an inversion.We now verify that ‖ · ‖ is indeed a norm on X ∼ . We haveandso {x n } ∼ {e}; also‖{x n } · {y n }‖ = lim n ‖x n y n ‖ ≤ lim n ‖x n ‖ + lim n ‖y n ‖,‖{x n }‖ = 0 iff lim n ‖x n ‖ = 0 iff x n → e,‖{x −1nWe note that if x n → R x, then}‖ = lim n ‖x −1n ‖ = lim n ‖x n ‖ = ‖{x n }‖.‖{x n } · {x −1 }‖ = lim n ‖x n x −1 ‖ = 0,so that {x n } → R {x} and hence the map x → {x n } where x n ≡ x isometrically embedsX into X ∼ . This far X ∼ is a normed group. Say that {x n } is d-regular if d(x n , x m ) ≤ 2 −nfor m ≥ n. If {{x m n } n } m is ˜d R -regular with each {x m n } also d R -regular, put y n = x n n. Then{y n } is the limit of {{x m n } n } m .Notice that if {w m n } → e, then without loss of generality w n n is null, so we haveand so X ∼ also satisfies (C-adm).x n nw n n(x n n) −1 → e X ,Remarks. 1. The definition of ˜X requires sequences to be bi-Cauchy to achieve bi-Cauchy completeness. Compare this two-sided condition to that of Prop. 3.13 whichuses bi-uniformly continuous functions, and also [BePe] Prop 1.1, where in the context ofAuth(X) with the weak refinement topology (that defined in Th. 2.12, as opposed to thatof Th. 3.19, where there is an abelian norm), the two-sided assumptions lim n f n = f ∈ X Xand lim n f −1 = g ∈ X X (limits in the supremum metric) yield g = f −1 ∈ Auth(X). (Onthis last point see also Lemma 1 of [Ost-Joint].)2. If X is complete under d R there is no guarantee that X is closed under products ofCauchy sequences, so Th. 3.35 does not characterize (C-adm).We now consider the impact of automatic continuity. Our first result captures theeffect on automorphisms of the result, due to Darboux [Dar], that an additive functionwhich is locally bounded is continuous.Definition. Say that a group is Darboux-normed if there are constants κ n with κ n → ∞associated with the group-norm such that for all elements z of the groupor equivalentlyκ n ‖z‖ ≤ ‖z n ‖,‖z 1/n ‖ ≤ 1κ n‖z‖.Thus z 1/n → e; a related condition was considered by McShane in [McSh] (cf. theEberlein-McShane Theorem, Th. 10.1).