Normed versus topological groups: Dichotomy and duality

56 N. H. Bingham and A. J. OstaszewskiTheorem 3.39. A Darboux-normed group is a topological group.Proof. Fix x. By Theorem 3.4 we must show that γ x is continuous at e. By Darboux’stheorem (Th. 11.22), it suffices to show that γ x is bounded in some ball B 1/n (e). Supposenot: then there is w n ∈ B ε(n) (e) with ε(n) = 2 −n and‖γ x (w n )‖ ≥ n.Thus w n is null. We may solve the equation w n = z n−1 zn−1by taking z 0 = e and inductivelyIndeed z n is null, sincez n = w −1nApplying the triangle inequality twice,as ‖z n ‖ ≤ 1. So for all n, we havez n−1 = wn−1 wn−1 −1 ...w−1 1 .‖z n ‖ ≤ 2 −(1+2+...+n) → 0.d(xz n , xz m ) ≤ d(xz n , e) + d(e, xz m )= ‖xz n ‖ + ‖xz m ‖ ≤ 2(‖x‖ + 1),for n = 1, 2, ... with z n null,‖xw n+1 x −1 ‖ = ‖x n z n z −1n−1 x−1 ‖ ≤ d(x n z n , x n−1 z n−1 ) ≤ 2(‖x‖ + 1).This contradicts the unboundedness of ‖γ x (w n )‖.Two more results on the effects of automatic continuity both come from the Banach-Mehdi Theorem on Homomorphism Continuity (Th. 11.11) or its generalization, theSouslin Graph Theorem (Th. 11.12), both of which belong properly to a later circle ofideas considered in Section 11 and employ the Baire property.Theorem 3.40. For X a topologically complete, separable, normed group, if each automorphismγ g (x) = gxg −1 is Baire, then X is a topological group.Proof. We work under d R . Fix g. As X is separable and γ g Baire, γ g is Baire-continuous(Th. 11.8) and so by the Banach-Mehdi Theorem (Th. 11.11) is continuous. As g isarbitrary, we deduce from Th. 3.4 that X is topological.Remark. Here by assumption (X, d R ) is a Polish space. In such a context, abandoningthe Axiom of Choice, one may consistently assume that all functions are Baire and sothat all topologically complete separable normed groups are topological. (See the modelsof set theory due to Solovay [So] and to Shelah [She].)Theorem 3.41 (On Borel/analytic inversion). For X a topologically complete, separablenormed group, if the inversion x → x −1 regarded as a map from (X, d R ) to (X, d R ) is aBorel function, or more generally has an analytic graph, then X is a topological group.