Normed versus topological groups: Dichotomy and duality

Normed groups 49Remark. On the matter of continuity a theorem of Mueller ([Mue, Th. 3] , see Th. 4.6below) asserts that in a locally compact group a subadditive p satisfyingis continuous almost everywhere.lim inf x→e (lim sup y→x p(y)) ≤ 03.3. Cauchy Dichotomy. In this section we demonstrate the impact on the structureof normed groups of the classic Cauchy dichotomy of homomorphisms; conjugacy is ahomomorphism so, in an appropriate setting, it is either continuous or highly discontinuousand so pathological, as mentioned in the Introduction. Thus, since conjugacy is atthe heart of normed groups, normed groups are in turn either topological or pathological(see e.g. Theorems 3.39 - 3.41 below, inspired by automatic continuity). The key here isDarboux’s classical result on automatic continuity [Dar], that an additive function on thereals is continuous if it is locally bounded, and its later weakening via ‘regularity’ to theBaire property (for which see below) or Haar-measurability in a locally compact context.Our aim here is to develop connections between continuity of automorphisms and threeareas: completeness, the Baire property and ‘boundedness’ of automorphisms. In respectof completeness, the findings (see e.g. Th. 3.37) are in keeping with tradition as exemplifiedby [Kel, Problem 6.Q]; the Baire property yields normed groups as ‘automaticallytopological’ (see the earlier cited theorems); boundedness is far more illuminating – inparticular we see that if the KBD holds in the right norm topology with a left-shift inthe standard form (i.e., as in Th.1.1, with tz m in lieu of t +z m ), as opposed to the specialform of Th.1.2 (yet to be established in Section 5), then the normed group is topological.In view of the importance of the Baire property to subsequent arguments, we recallthat a set is meagre if it is a countable union of nowhere dense sets, a set is Baireif it is open modulo a meagre set, or equivalently if it is closed modulo a meagre set(cf. Engelking [Eng] especially p.198 Section 4.9 and Exercises 3.9.J, although we prefer‘meagre’ to ‘of first category’ ). For examples, see Section 11.Definition. Noting that d L (x n , x m ) = d R (x −1n , x −1m ), call a sequence {x n } bi-Cauchy(or two-sided Cauchy) if {x n } is both d R - and d L -Cauchy, i.e. both {x n } and {x −1n }are d R -Cauchy sequences. Thus a sequence is bi-Cauchy iff it is d S -Cauchy, where d S =max{d R , d L } is the symmetrization metric. Recall that d S induces a norm, the originatingnorm, as ‖x‖ S := d S (x, e) = ‖x‖, but does not in general induce the same topology asd R .Discontinuity of automorphisms may be approached through bi-Cauchy sequences.Indeed, if a normed group is not topological, then by Th. 3.4 there are a null sequencez n , a point t, and ε > 0 such thatε ≤ lim n ‖tz n t −1 ‖;then d R (tz n , t) ≥ ε, and tz n is prevented from converging to t. Thus one asks whether{tz n } has a convergent subsequence {tz m } m∈M (such a sequence would be bi-Cauchy, ast −1 } is Cauchy, since for w n null w n x → R x, for any x).{z −1n