Normed versus topological groups: Dichotomy and duality

Normed groups 105By the Category Embedding Theorem (Th. 6.1), for some λ ∈ T ′ = u −1 T, we have withλ = u −1 t and for infinitely many nsou −1 v n : = ˜z n z n λ ∈ T ′ = u −1 T,u˜z n z n λ = v n ∈ T, or u˜z n z n uu −1 λ = v n ∈ T,u 2 nu −1 λ = v n ∈ T, or u 2 n = v n λ −1 u = v n v(with v = λ −1 u = t −1 u 2 ∈ T −1 u 2 ).As for the remaining assertion, note that u −1n → u −1 , vn−1 ∈ T −1 andu −2n= v −1 v −1n .Thus noting that T −1 is non-meagre (since inversion is a homeomorphism) and replacingT −1 by T we obtain the required assertion by a right-averaging translator.Note the connection between the norms of the null sequences is only by way of theconjugate metrics:‖z n ‖ = d(e, u n u −1 ) = d(u, u n ), and ‖˜z n ‖ = d(e, u −1 u n ) = d(u −1n , u −1 ) = ˜d(u n , u).Whilst we may make no comparisons between them, both norms nevertheless convergeto zero.Definitions. For G, H normed groups, we say that f : G → H is locally Lipschitz at gif, for some neighbourhood N g of g and for some constants K g and all x, y in N g ,∣ ∣ ∣f(x)f(y) −1∣ ∣ ∣ ∣H≤ K g ‖xy −1 ‖ G .We say that f : G → H is locally bi-Lipschitz at g if, for some neighbourhood N g of gand for some positive constants K g , κ g , and all x, y in N g ,κ g ‖xy −1 ‖ G ≤ ∣ ∣ ∣ ∣f(x)f(y) −1∣ ∣ ∣ ∣H≤ K g ‖xy −1 ‖ G .If f : G → H is invertible, this asserts that both f and its inverse f −1 are locally Lipschitzat g and f(g) respectively.We say that the norm on G is n-Lipschitz if the function f n (x) := x n from G to G islocally Lipschitz at all g ≠ e, i.e. for each there is a neighbourhood N g of g and positiveconstants κ g , K g so thatκ g ‖xy −1 ‖ G ≤ ∣ ∣ ∣ ∣ x n y −n∣ ∣ ∣ ∣G≤ K g ‖xy −1 ‖ G .In an abelian context the power function is a homomorphism; we note that [HJ, p.381] refers to a semigroup being modular when each f n (defined as above) is an injectivehomomorphism. The condition on the right with K = n is automatic, and so one needrequire only that for some positive constant κκ‖g‖ ≤ ‖g n ‖.Note that, in the general context of an n-Lipschitz norm, if x n = y n , then as (x n y −n ) = e,we have κ g ‖xy −1 ‖ G ≤ ||x n y −n || G= ‖e‖ = 0, and so ‖xy −1 ‖ G = 0, i.e. the power function