Normed versus topological groups: Dichotomy and duality

8 N. H. Bingham and A. J. OstaszewskiwhereB i ε(x) := {y ∈ X : d X i (x, y) < ε}.2. The strong ∆-refinement topology on X is defined by reference to the local base at xobtained by full intersections of ε-balls about x :⋂(Str)Clearly⋂d∈∆ Bd ε (x).d∈∆ Bd ε (x) ⊂ ⋂ i∈F Bi ε(x), for F finite,hence the name. We will usually be concerned with a family ∆ of conjugate metrics. Wenote the following, which is immediate from the definition. (For (ii) see the special casein [dGMc, Lemma 2.1], [Ru, Ch. I 1.38(c)], or [Eng, Th. 4.2.2 p. 259], which uses a sumin place of a supremum, and identify X with the diagonal of ∏ d∈∆(X, d); see also [GJ,Ch. 15].)Proposition 2.1. (i) The strong ∆-refinement topology is generated by the supremummetricd X ∆(x, y) = sup{d X i (x, y) : i ∈ I}.(ii) For ∆ a countable family of metrics indexed by I = N, the weak ∆-refinement topologyis generated by the weighted-supremum metricd X ∆(x, y) = sup i∈I 2 1−i d X i (x, y)1 + d X i (x, y).This corresponds to the metric of first-difference in a product of discrete metric spaces,e.g. in the additive group Z Z . (That is, d X ∆ ({x i}, {y i }) = 2 −n(x,y) , where the two sequencesfirst differ at index i = n(x, y).)Examples B. 1. For X a group we may take ∆ = {d X zand if d X is right-invariantd X ∆(x, y) = sup{d X (zx, zy) : z ∈ X},‖x‖ ∆ = sup z ‖zxz −1 ‖.: z ∈ X} to obtain2. For X a topological group we may take ∆ = {d X h: h ∈ Auth(X)}, to obtaind X ∆(x, y) = sup{d X (h(x), h(y)) : h ∈ Auth(X)}.3. In the case A = Auth(X) we may take ∆ = {d A x : x ∈ X}, the evaluation pseudometrics,to obtaind A ∆(f, g) = sup x d A x (f, g) = sup x d X (f(x), g(x)),‖f‖ ∆ = sup x d A x (f, id X ) = sup x d X (f(x), x).In Proposition 2.12 we will show that the strong ∆-refinement topology restricted to thesubgroup H(X) := {f ∈ A : ‖f‖ ∆ < ∞} is the topology of uniform convergence. Theweak ∆-refinement topology here is just the topology of pointwise convergence.and