Normed versus topological groups: Dichotomy and duality

132 N. H. Bingham and A. J. OstaszewskiRemark. Alternatively, working in T r L (X) rather than in H u (X) and with d X R againright-invariant, since ξ x (λ y )(z) = λ y λ −1x (z)) = λ yx −1(z), we havepossibly infinite. Indeedsup w d H (ξ x (λ w ), ξ e (λ w )) = sup v d X v (e, x) = ‖x‖ X ∞,sup w d H (ξ x (λ w ), ξ y (λ w )) = sup w sup z d X R (ξ x (λ w )(z), ξ y (λ w )(z))= sup w sup z d X R (wx −1 z, wy −1 z) = sup w d X R (vxx −1 , vxy −1 )= sup v d X R (vy, vx) = sup v d X v (y, x).(Here we have written w = vx.)The refinement metric sup v d X (vy, vx) is left-invariant on the bounded elements (i.e.bounded under the corresponding norm ‖x‖ := sup{‖vxv −1 ‖ : v ∈ X}; cf. Proposition2.12). Of course, if d X were bi-invariant (both right- and left-invariant), we would havesup w d H (ξ x (λ w ), ξ y (λ w )) = d X (x, y).13. Divergence in the bounded subgroupIn earlier sections we made on occasion the assumption of a bounded norm. Here we areinterested in norms that are unbounded. For S a space and A a subgroup of Auth(S)equipped with the supremum norm, suppose ϕ : A × S → S is a continuous flow (seeLemma 3.8, for an instance). We will write α(s) := ϕ α (s) = ϕ(α, s). This is consistentwith A being a subgroup of Auth(S). As explained at the outset of Section 12, we havein mind two pairs (A, S), as follows.Example 1. Take S = X to be a normed topological group and A = T ⊆ H(X) to bea subgroup of automorphisms of X such that T is a topological group with supremummetricd T (t 1 , t 2 ) = sup x d X (t 1 (x), t 2 (x)),e.g. T = H u (X). Note that here e T = id X .Example 2. (A, S) = (Ξ, T ) = (X, T ). Here X is identified with its second dual Ξ (ofthe preceding section).Given a flow ϕ(t, x) on T × X, with T closed under translation, the action defined byϕ(ξ x , t) := ξ x −1(t)is continuous, hence a flow on Ξ × T, which is identified with X × T . Observe thatt(x) = ξ x −1(t)(e X ), i.e. projection onto the e X coordinate retrieves the T -flow ϕ. Here,for ξ = ξ x −1, writing x(t) for the translate of t, we haveξ(t) := ϕ ξ (t) = ϕ(ξ, t) = x(t),so that ϕ may be regarded as a X-flow on T. We now formalize the notion of a sequence