Normed versus topological groups: Dichotomy and duality

28 N. H. Bingham and A. J. OstaszewskiLemma 3.2. If inversion is right-to-right continuous, thenx → R a iff a −1 x → R e.Proof. For x → R a, we have d R (e, a −1 x) = d R (x −1 , a −1 ) → 0, assuming continuity.Conversely, for a −1 x → R e we have d R (a −1 x, e) → 0, i.e. d R (x −1 , a −1 ) → 0. So sinceinversion is assumed to be right-continuous and (x −1 ) −1 = x, etc, we have d R (x, a) → 0.We now expand this.Theorem 3.3. The following are equivalent:(i) inversion is right-to-right continuous,(ii) left-open sets are right-open,(iii) for each g the conjugacy γ g is right-to-right continuous at e, i.e. for every ε > 0there is δ > 0 such thatgB(δ)g −1 ⊂ B(ε),(iv) left-shifts are right-continuous.Proof. We show that (i)⇐⇒(ii)⇐⇒(iii)⇐⇒(iv).Assume (i). For any a and any ε > 0, by continuity of inversion at a, there is δ > 0 suchthat, for x with d R (x, a) < δ, we have d R (x −1 , a −1 ) < ε, i.e. d L (x, a) < ε. ThusB(δ)a = B R (a, δ) ⊂ B L (a, ε) = aB(ε),(incl)i.e. left-open sets are right-open, giving (ii). For the converse, we just reverse the lastargument. Let ε > 0. As a ∈ B L (a, ε) and B L (a, ε) is left open, it is right open and sothere is δ > 0 such thatB R (a, δ) ⊂ B L (a, ε).Thus for x with d R (x, a) < δ, we have d L (x, a) < ε, i.e. d R (x −1 , a −1 ) < ε, i.e. inversionis right-to-right continuous, giving (i).To show that (ii)⇐⇒(iii) note that the inclusion (incl) is equivalent toi.e. toa −1 B(δ)a ⊂ B(ε),γa−1 [B(δ)] ⊂ B(ε),that is, to the assertion that γ a (x) is continuous at x = e (and so continuous, by Lemma3.1). The property (iv) is equivalent to (iii) since the right-shift is right-continuous andγ a (x)a = λ a (x) is equivalent to γ a (x) = λ a (x)a −1 .We saw in the Birkhoff-Kakutani Theorem (Th. 2.19) that metrizable topologicalgroups are normable (equivalently, have a right-invariant metric); we now formulate aconverse, showing when the right-invariant metric derived from a group-norm equips itsgroup with a topological group structure. As this is a characterization of metric topologicalgroups, we will henceforth refer to them synonymously as normed topological groups.