Normed versus topological groups: Dichotomy and duality

Normed groups 61Proposition 3.46 (Montgomery’s Uniformity Lemma, [Mon2, Lemma]). For a normedgroupX under its right norm topology, Baire in this topology, and ε > 0, if Ω(ε) ∩ U isnon-meagre for some open set U, then there are δ = δ(ε) > 0 and an open V ⊂ U suchthat V ⊆ Λ δ (ε), i.e., d R (t, tz) ≤ ε for all ‖z‖ ≤ δ and t ∈ V.In particular, if in an open set U the oscillation is less than ε at points of a non-meagreset, then it is at most ε at all points of some non-empty open subset of U.Proof. As Ω δ (ε) ⊆ Λ δ (ε) (with d = d R ), we haveΩ(ε) ∩ U ⊂ ⋃ 1/δ∈N Λ δ(ε) ∩ U.So if U ∩ Ω(ε) is non-meagre, then U ∩ Λ δ (ε) is non-meagre for some δ > 0 and so, byBaire’s Theorem, dense in some open V with clV ⊂ H. But Λ δ (ε) is closed, so V ⊂ Λ δ (ε).Thus d(t, tz) ≤ ε for all ‖z‖ ≤ δ and t ∈ V.Proposition 3.47 (Montgomery’s Joint Continuity Theorem, [Mon2, Th. 1]). Let X bea normed group, locally complete in the right norm topology, and W a non-empty open setW . If Ω(ε) ∩ W is non-meagre for each ε > 0, then there is w ∈ W with γ w continuous.So if Ω(ε) ∩ U is non-meagre for each ε > 0 and each non-empty U ⊆ W , then W ∩ Z Γis dense in W.In particular, if Ω(ε) ∩ U is non-meagre for each ε > 0 and every open set U, then X isa topological group.More generally, if for some open W and all ε > 0 the set Ω(ε) ∩ W is non-meagre in W,and X is separable and connected, then X is a topological group.Proof. Working in the right topology, and by Prop. 3.46 taking successively ε(n) = 2 −nfor ε, we may choose inductively δ(n) and open sets U n with U n+1 ⊆ U n such thatU n+1 ⊆ Λ δ(n) (ε(n)). So if w ∈ ⋂ U n , then for each n we have ω δ(n) (w) ≤ ε(n), so thatω(w) = 0.The final assertion follows by Prop. 3.43, since now the centre Z Γ is dense in the space.The preceding result, already a sharpening of Montgomery’s original result, says thatif X is not a topological group then the oscillation is bounded away from zero on a comeagreset. But we can improve on this. It will be convenient (cf. Th. 3.48 below) tomake the followingDefinition. Working in the right norm topology (X, d R ), call t an ε-shifting point (onthe left) if there is δ > 0 such that for ‖z‖ ≤ δd R (t, tz) < ε,equivalently, in oscillation function terms, ω δ (t) ≤ ε (since ‖tzt −1 ‖ ≤ ε for ‖z‖ ≤ δ).