Normed versus topological groups: Dichotomy and duality

108 N. H. Bingham and A. J. Ostaszewskias the group is abelian. Sof(u) ≤ 1 2 f(yx 0) + 1 2 f(z) ≤ 1 2 M + 1 2 f(z t).That is, 1 2 (M + f(z t)) is an upper bound for f in B δ/2 (x 0 ).Case (ii) The general case. As before, suppose f is bounded above in B := B δ (x 0 ) byM,and let t be a given a fixed point; put z = z t := x −10 t2 so that t 2 = x 0 z.For this fixed t the mapping y → α(y) := ytyt −1 y −2 is continuous (cf. Th. 3. 7 oncommutators) with α(e) = e, so α(y) is o(y) as ‖y‖ → 0. Nowsts = [stst −1 s −2 ]s 2 t = α(s)s 2 t,and we may suppose that, for some η < δ/2, we have ‖α(s)‖ < δ/2, for ‖s‖ < η. Notethatstst = α(s)s 2 t 2 .Consider any u ∈ B r (t) with r = min{η, δ/2}. Write u = st with ‖s‖ < r ≤ δ/2. Now puty = s 2 . Then ‖y‖ = ‖s 2 ‖ ≤ 2‖s‖ < δ and ‖o(s)y‖ ≤ η +δ/2 < δ. Hence o(s)yx 0 ∈ B δ (x 0 ).NowHence, by convexity,u 2 = stst = α(s)s 2 t 2 = α(s)yx 0 z.f(u) ≤ 1 2 f(o(s)yx 0) + 1 2 f(z) ≤ 1 2 M + 1 2 f(z t).As an immediate corollary of the last theorem and the Bernstein-Doetsch Theorem(Th. 10.6) we have the following result.Theorem 10.8 (Dichotomy Theorem for convex functions – [Kucz, p.147]). In a normedtopological group, for 1 2-convex f (so in particular for additive f) either f is continuouseverywhere, or it is discontinuous everywhere.The definition below requires continuity of ‘square-rooting’ – taken in the form of analgebraic closure property of degree 2 in a group G, expressed as the solvability of certain‘quadratic equations’ over the group. Its status is clarified later by reference to Bartle’sInverse Function Theorem. We recall that a group is n-divisible if x n g = e is soluble foreach g ∈ G. (In the absence of algebraic closure of any degree an extension of G may beconstructed in which these equations are solvable – see for instance Levin [Lev].)Definition. We say that the normed group G is locally convex at λ = t 2 if for any ε > 0there is δ > 0 such that for all g with ‖g‖ < ε the equationxtxt = gt 2 ,equivalently xtxt −1 = g, has its solutions satisfying ‖x‖ < δ.