Normed versus topological groups: Dichotomy and duality

50 N. H. Bingham and A. J. OstaszewskiThis approach suggests another: if y = lim M tz m , the distance d R (y, t) measures thediscontinuity of λ t in the corresponding direction {z m } M , since‖yt −1 ‖ = d R (y, t) = lim M d R (tz m , t) = lim M ‖γ t (z m )‖,leading to a study of the properties of the oscillation (at e X ) of γ t as t varies over thegroup, which we address later in this subsection. (Note that here ‖y‖ = ‖t‖, so themaximum dispersion by a left-shift of the null sequence, away from where the right-shifttakes it, is ‖yt −1 ‖ ≤ 2‖t‖; it is this that the oscillation measures.) We shall see laterthat if a normed group is not topological than the oscillation is bounded away from zeroon a non-empty open set, suggesting a considerable amount (in topological terms) ofpathology.Returning to the bi-Cauchy approach, in order to draw our work closer to the separatecontinuity literature (esp. Bouziad [Bou2]), we restate it in terms of the following notion ofcontinuity due to Fuller [Ful] (in his study of the preservation of compactness), adaptedhere from nets to sequences, because of our metric context. (Here one is reminded ofcompact operators – cf. [Ru, 4.16])Definition. A function f between metric spaces is said to be subcontinuous at x if foreach sequence x n with limit x, the sequence f(x n ) has a convergent subsequence.Thus for f(x) = λ t (x) = tx, with t fixed, λ t is subcontinuous under d R at e iff for eachnull z n there exists a convergent subsequence {tz n } n∈M . We note that λ t is subcontinuousunder d R at e iff it is subcontinuous at some/all points x (since tz m x → R yx iff tz m → R ydown the same subsequence M, and x n → R x iff z n := x n x −1 → e so that z n x → R x.)One criterion for subcontinuity is provided by a form of the Heine-Borel Theorem,which motivates a later definition.Proposition 3.32 (cf. [Ost-Mn, Prop. 2.8]). Suppose that Y = {y n : n = 1, 2, ..} is aninfinite subset of a normed group X. Then Y contains a subsequence {y n(k) } which iseither d R -Cauchy or is uniformly separated (i.e. for some m satisfies d R (y n(k) , y n(h) ) ≥1/m, for all h, k). In particular, if X is locally compact, z n is null, and the ball B ‖x‖ (e)is precompact, then y n = xz n contains a d R -Cauchy sequence.Proof. We may assume without loss of generality that y n is injective and so identifyY with N. Define a colouring M on N by setting M(h, k) = m iff m is the smallestinteger such that d R (y h , y k ) ≥ 1/m. If an infinite subset I of N is monochromatic withcolour n, then {y i : i ∈ I} is a discrete subset in X. Now partition N 3 by putting{u, v, w} in the cells C < , C = , C > according as M(u, v) < M(v, w), M(u, v) = M(v, w), orM(u, v) > M(v, w). By Ramsey’s Theorem (see e.g. [GRS, Ch.1]), one cell contains aninfinite set I 3 . As C > cannot contain an infinite (descending) sequence, the infinite subsetis either in C = , when {y i : i ∈ I} is uniformly separated, or in C < , when {y i : i ∈ I} is ad R -Cauchy sequence.