Normed versus topological groups: Dichotomy and duality

134 N. H. Bingham and A. J. OstaszewskiExamples. In R we may consider ϕ n (t) = t + x n where x n → ∞. In a more generalcontext, a natural example of a uniformly divergent sequence of homeomorphisms isagain provided by a flow parametrized by discrete time from a source to infinity. Ifϕ : N × X → X is a flow and ϕ n (x) = ϕ(n, x), then, for each x, the orbit {ϕ n (x) :n = 1, 2, ...} is the image of the divergent real sequence {y n (x) : n = 1, 2, ...}, wherey n (x) := d(ϕ n (x), x) ≥ d ∗ (ϕ n , id).Remark. Our aim is to offer analogues of the topological vector space characterizationof boundedness: for a bounded sequence of vectors {x n } and scalars α n → 0 ([Ru, cf. Th.1.30]), α n x n → 0. But here α n x n is interpreted in the spirit of duality as α n (x n ) withthe homeomorphisms α n converging to the identity.Examples. 1. Evidently, if S = X, the pointwise definition reduces to functional divergencein H(X) defined pointwise:d X (α n (x), x) → ∞.The uniform version corresponds to divergence in the supremum metric in H(X).2. If S = T and A = X = Ξ, we have, by the Quasi-isometric Duality Theorem (Th.12.7), thatd T (ξ x(n) (t), ξ e (t)) → ∞iffd X (x n , e X ) → ∞,and the assertion is ordinary divergence in X. Sincethe uniform version also asserts thatd Ξ (ξ x(n) , ξ e ) = d X (x n , e X ),d X (x n , e X ) → ∞.Recall that ξ x (s)(z) = s(λ −1x (z)) = s(x −1 z), so the interpretation of Ξ as having theaction of X on T was determined byOne may writeϕ(ξ x , t) = ξ x −1(t)(e) = t(x).ξ x(n) (t) = t(x n ).When interpreting ξ x(n) as x n in X acting on t, note thatd X (x n , e X ) ≤ d X (x n , t(x n )) + d X (t(x n ), e X ) ≤ ‖t‖ + d X (t(x n ), e X ),so, as expected, the divergence of x n implies the divergence of t(x n ).The next definition extends our earlier one from sequential to continuous limits.