Normed versus topological groups: Dichotomy and duality

2 N. H. Bingham and A. J. Ostaszewski(in the study of products of regularly varying functions with range a normed group) – see[BOst-TRIII]. In a normed group topological completeness yields a powerful Shift Theorem,generalizing the following theorem on the reals about shift embedding of sequences(due in a weak form to Kestelman and in a Lebesgue-measure setting to Borwein andDitor). We say generically all to mean ‘off a meagre/null set’, according to whether thecontext is (Baire) category, where we also say quasi all, or (Lebesgue, or more generallyHaar) measure, where we say almost all.Theorem 1.1 (Kestelman-Borwein-Ditor Theorem, KBD). Let {z n } → 0 be a nullsequence of reals. If T is measurable and non-null, or Baire and non-meagre, then forgenerically all t ∈ T there is an infinite set M t such that{t + z m : m ∈ M t } ⊆ T.A stronger form still is derived in [BOst-Funct] (the Generic Reflection Theorem);see also [BOst-StOstr] Section 3.1 Note 3, [BOst-LBI] Section 3.1 Note 1. For proofs seethe original papers [Kes] and [BoDi]; for a unified treatment in the real-variable casesee [BOst-Funct]. Applications of shift embedding are implicit in Banach [Ban-Eq] andexplicit though not by name in Banach [Ban-T] in the proofs that a measurable/Baireadditive function is continuous (see the commentary by Henryk Fast loc. cit. p. 314for various one-way implications among related results). The present paper is motivatedprecisely by normed groups being the natural setting for generalizations of the KBDTheorem and its numerous important applications (initially noticed in the Uniform ConvergenceTheorem of the theory of regular variation). Normed groups, as we will see,are subject to a dichotomy centered on automatic continuity (for background see Section3.3 and Section 11), as to whether or not inner automorphisms x → gxg −1 are continuous:normed groups are thus either topological groups or pathological groups. That is,a smidgen of regularity tips the normed group over to a topological group. We are thusmostly concerned with the former; but even so in general, in the presence of completeness,they support a generalization of KBD from which one may derive a Squared PettisTheorem (that (AA −1 ) 2 , for A Baire non-meagre, has the identity as an interior point,Th.5.8); that in turn guarantees in the category of normed groups the Banach-MehdiContinuity Theorem for Baire-continuous homomorphisms (Th 11.10), the Baire HomomorphismTheorem (Th.11.11) and the Souslin Graph Theorem (Th. 11.12). The originof the squaring is the following first of several generalizations of KBD (cf. Th. 5.1).Theorem 1.2 (Kestelman-Borwein-Ditor Theorem – Normed Groups). In a topologicallycomplete normed group X, if {z n } → e X (a null sequence converging to the identity), Tis Baire and non-meagre under the right norm topology, then there are t, t m ∈ T and aninfinite set M t such that{tt −1m z m t m : m ∈ M t } ⊆ T.