Normed versus topological groups: Dichotomy and duality

Normed groups 51As for the conclusion, the set B ‖x‖+ε (e) has compact closure for some ε > 0. But ‖xz n ‖ ≤‖x‖ + ‖z n ‖, so for large enough n the points xz n lie in the compact set B ‖x‖+ε (e), hencecontain a convergent subsequence.Our focus on metric completeness is needed in part to supply background to assumptionsin Section 5 (e.g. Th. 5.1). We employ definitions inspired by weakening theadmissibility condition (adm). We recall from Th. 2.15 that a normed group with (adm) isa topological group, more in fact: a Klee group, as it has an equivalent abelian norm. Wewill see that the property of (only) being a topological group is equivalent to a weakenedadmissibility property; a second (less weakened) notion of admissibility – the Cauchyadmissibilityproperty – ensures that (X, d R ) has a group completion. This motivates theuse in Section 5 of the weaker property still that (X, d R ) is topologically complete, i.e.there is a complete metric d on X equivalent to d R .Definitions. 1. Say that the normed group satisfies the weak-admissibility condition,or (W-adm) for short, if for every convergent {x n } and null {w n }x n w n x −1n → e, as n → ∞. (W-adm)Note that the (W-adm) condition has a reformulation as the joint continuity of the leftcommutator [x, y] L , at (x, y) = (w, e), when the convergent sequence {x n } has limit x;indeedx n w n x −1n= x n w n x −1n wn −1 w n = [x n , w n ] L w n .Likewise, if the sequence {x −1n } has limit x −1 , then one can writex n w n x −1n= x n w n x −1n w n wn−1= [x −1n, wn−1 ] R wn −1 .2. Say that the normed group satisfies the Cauchy-admissibility condition, or (C-adm)for short, if for every Cauchy {x n } and null {w n }x n w n x −1n → e, as n → ∞. (C-adm)In what follows, we have some flexibility as to when x n is a Cauchy sequence. Oneinterpretation is that x n is d R -Cauchy, i.e. ‖x n x −1m ‖ = d R (x n , x m ) → 0. The other is thatx n is d L -Cauchy; but then y m = x −1n is d R -Cauchy and we havex n w n x −1n= y −1n w n y n → e.The distinction is only in the positioning of the inverse; hence in arguments, as below,which do not appeal to continuity of inversion, either format will do.Lemma 3.33. In a normed group the condition (C-adm) is equivalent to the followinguniformity condition holding for all {x n } Cauchy:for each ε > 0 there is δ > 0 and N such that for all n > N and all ‖w‖ < δ‖x n wx −1n ‖ < ε.