Normed versus topological groups: Dichotomy and duality

40 N. H. Bingham and A. J. OstaszewskiRemark. To see the need for the refinement norm in verifying continuity of compositionin H(X), we work with metrics and recall the permutation metric d X g (x, y) :=d X (g(x), g(y)). Recall also that the metric defined by the norm ‖f‖ g is the supremummetric ˆd g on H(X) arising from d g on X. Indeedd g (h ′ , h) = ‖h ′ h −1 ‖ g = sup z d X (gh ′ h −1 g −1 (z), z) = sup x d X (g(h ′ (x)), g(h(x)))= sup x d X g (h ′ (x)), h(x)).Since, as in Proposition 2.13,ˆd g (F 1 G 1 , F G) ≤ ˆd g (F 1 , F ) + ˆd gF (G 1 , G) ≤ ˆd ∞ (F 1 , F ) + ˆd ∞ (G 1 , G),we may conclude thatˆd ∞ (F 1 G 1 , F G) ≤ ˆd ∞ (F 1 , F ) + ˆd ∞ (G 1 , G).This reconfirms that composition is continuous. When g = e, the term ˆd F arises aboveand places conditions on how ‘uniformly’ close G 1 needs to be to G (as in Th. 3.13).For these reasons we find ourselves mostly concerned with H u (X).3.2. Lipschitz-normed groups. Below we weaken the Klee property, characterizedby the condition ‖gxg −1 ‖ ≤ ‖x‖, by considering instead the existence of a real-valuedfunction g → M g such that‖gxg −1 ‖ ≤ M g ‖x‖, for all x.This will be of use in the development of duality in Section 12 and partly in the considerationof the oscillation of a normed group in Section 3.3.Remark. Under these circumstances, on writing xy −1 for x and with d X the rightinvariantmetric defined by the norm, one hasd X (gxg −1 , gyg −1 ) = d X (gx, gy) ≤ M g d X (x, y),so that the inner-automorphism γ g is uniformly continuous (and a homeomorphism).Moreover, M g is related to the Lipschitz-1 norms ‖g‖ 1 and ‖γ g ‖ 1 , whered X (gx, gy)‖g‖ 1 := sup x≠yd X (x, y) , and ‖γ d X (gxg −1 , gyg −1 )g‖ 1 := sup x≠yd X ,(x, y)cf. [Ru, Ch. I, Exercise 22]. This motivates the following terminology.Definitions. 1. Say that an automorphism f : G → G of a normed group has theLipschitz property if there is M > 0 such that‖f(x)‖ ≤ M‖x‖, for all x ∈ G.(Lip)2. Say that a group-norm has the Lipschitz property , or that the group is Lipschitznormed,if each continuous automorphism has the Lipschitz property under the groupnorm.Definitions. 1. Recall from the definitions of Section 2 that a group G is infinitelydivisible if for each x ∈ G and n ∈ N there is some ξ ∈ G with x = ξ n . We may writeξ = x 1/n (without implying uniqueness).