Normed versus topological groups: Dichotomy and duality

16 N. H. Bingham and A. J. OstaszewskiProposition 2.11 (Group-norm properties in H(X)). If ‖h‖ = ‖h‖ H , then ‖ · ‖ is agroup-norm: that is, for h, h ′ ∈ H(X),‖h‖ = 0 iff h = e, ‖h ◦ h ′ ‖ ≤ ‖h‖ + ‖h ′ ‖ and ‖h‖ = ‖h −1 ‖.Proof. Evidently ˆd(h, id X ) = sup x∈X d(h(x), x) = 0 iff h(x) = id X . We have‖h‖ = ˆd(h, id X ) = sup x∈X d(h(x), x) = sup y∈X d(y, h −1 (y)) = ‖h −1 ‖.Next note thatˆd(id X , h ◦ h ′ ) = sup x∈X d(hh ′ (x), x) = sup y∈X d(h(y), h ′−1 (y)) = ˆd(h, h ′−1 ). (right-inv)Butˆd(h, h ′ ) = sup x∈X d(h(x), h ′ (x)) ≤ sup x∈X [d(h(x), x) + d(x, h ′ (x))]≤ ˆd(h, id) + ˆd(h ′ , id) < ∞.Theorem 2.12. The set H(X) of bounded self-homeomorphisms of a metric group X isa group under composition, metrized by the right-invariant supremum metric ˆd X .Proof. The identity, id X , is bounded. For right-invariance (cf. (right-inv)),ˆd(g ◦ h, g ′ ◦ h) = sup x∈X d(g(h(x)), g ′ (h(x)) = sup y∈X d(g(y), g ′ (y)) = ˆd(g, g ′ ).Theorem 2.13 ([BePe, Ch. IV Th 1.1]). Let d be a bounded metric on X. As a groupunder composition, A = Auth(X) is a topological group under the weak ∆-refinementtopology for ∆ := { ˆd π : π ∈ A}.Proof. To prove continuity of inversion at F, write H = F −1 and for any x put y =f −1 (x). Thend π (f −1 (x), F −1 (x)) = d π (H(F (y)), H(f(y))) = d πH (F (y), f(y)),and soˆd π (f −1 , F −1 ) = sup x d π (f −1 (x), F −1 (x)) = sup y d πH (F (y), f(y)) = ˆd πH (f, F ).Thus f −1 is in any ˆd π neighbourhood of F −1 provided f is in any ˆd πH neighbourhood ofF.As for continuity of composition at F, G, we have for fixed x thatHenced π (f(g(x)), F (G(x))) ≤ d π (f(g(x)), F (g(x))) + d π (F (g(x)), F (G(x)))= d π (f(g(x)), F (g(x)) + d πF (g(x), G(x))≤ ˆd π (f, F ) + ˆd πF (g, G).ˆd π (fg, F G) ≤ ˆd π (f, F ) + ˆd πF (g, G),