Normed versus topological groups: Dichotomy and duality

94 N. H. Bingham and A. J. Ostaszewskimay be given priority. See in particular [BGT, p. 5,11] and [BOst-LBII].3. The Kestelman-Borwein-Ditor Theorem inspires the following definitions, which wewill find useful in the next section.Definitions. In a topological group G, following [BOst-FRV], call a set T subuniversal,or null-shift-precompact as in the more recent paper [BOst-StOstr], if for any null sequencez n → e G there is t ∈ G and infinite M t such that{tz m : m ∈ M t } ⊂ T.Call a set T generically subuniversal ([BOst-FRV]), or null-shift-compact (cf. [BOst-StOstr]),if for any null sequence z n → e G there is t ∈ G and infinite M t such that{tz m : m ∈ M t } ⊂ T and t ∈ T.Thus the Kestelman-Borwein-Ditor Theorem asserts that a set T which is Baire nonmeagre,or measurable non-null, is (generically) subuniversal. The term subuniversal iscoined from Kestelman’s definition of a set being ‘universal for null sequences’ ([Kes,Th. 2]) , which required M t above to be co-finite rather than infinite. By Theorem 6.7(Shift-compactness Theorem), a generically subuniversal (null-shift-compact) subset ofa normed group is shift-compact. (The definition of ‘shift-compact’ refers to arbitrarysequences – see Section 6.)Our final results follow from the First Generalized KBD Theorem (Th. 7.6 above)and are motivated by the literature of extended regular variation in which one assumesonly that for a function h : R + → R +h ∗ (u) := lim sup ‖x‖→∞ h(ux)h(x) −1is finite on a ‘large enough’ domain set (see [BOst-RVWL], [BGT] Ch. 2,3 for the classicalcontext of R ∗ +). We need the following definitions generalizing their R counterparts (in[BOst-RVWL]) to the normed group context.Definitions. 1. Say that NT ∗ ({T k }) holds, in words No Trumps holds generically, iffor any null sequence z n → e X there is k ∈ ω and an infinite M such that{tz m : m ∈ M} ⊂ T k and t ∈ T k .For the definition of NT see [BOst-FRV], [BOst-LBI] where bounded, rather than null,sequences z n appear and the location of the translator t need not be in T k . [Of courseNT ∗ ({T k : k ∈ ω}) implies NT({T k : k ∈ ω}).]2. For X a normed group, h : T → Y or R + , with Y a normed group, put according tocontext:h ∗ (u) := lim sup ‖x‖→∞ h(ux)h(x) −1 or h ∗ Y (u) := lim sup ‖x‖→∞ ‖h(ux)h(x) −1 ‖ Y .