Normed versus topological groups: Dichotomy and duality

80 N. H. Bingham and A. J. OstaszewskiIn the theorem below D is dense under d R ; this means that D −1 , and so each xD −1 ,are dense under d L . The surprise is that, for quasi all x, xD −1 is also dense under d R .Theorem 5.10 (Generic Density Theorem, [HJ, Th.2.3.7]). Let X be Baire under its rightnorm topology with a density-preserving norm. For A co-meagre in X and D countableand dense under d R{x ∈ A : (xD −1 ) ∩ A is dense in X}is co-meagre in X.Proof. For each x, the set Ax is co-meagre as ρ x (t) = tx is a homeomorphism. HenceB = A ∩ ⋂ d∈D Adis co-meagre, as D is countable. Thus for x ∈ B and d ∈ D we have xd −1 ∈ A. Now let Vbe open with a ∈ B r (a) = B r (e)a ⊂ V. Let x ∈ X. We claim that there is d ∈ D such thatx ∈ B r (a)d. By assumption aD is dense, so there is ad ∈ B r (x) = B r (e)x. Put ad = zxwith ‖z‖ < r. Then x = z −1 ad ∈ B r (a)d, as claimed. Thus v := xd −1 = z −1 a ∈ V andso for x ∈ B we havev = xd −1 ∈ (xD −1 ) ∩ A ∩ V.That is, (xD −1 ) ∩ A is dense in X, for x in the co-meagre set B.6. Steinhaus theory and DichotomyIf ψ n converges to the identity, then, for large n, each ψ n is almost an isometry. Indeed,as we shall see in Section 12, by Proposition 12.5, we haved(x, y) − 2‖ψ n ‖ ≤ d(ψ n (x), ψ n (y)) ≤ d(x, y) + 2‖ψ n ‖.This motivates our next result; we need to recall a definition and the Category EmbeddingTheorem from [BOst-LBII], whose proof we reproduce here for completeness.Definition (Weak category convergence). A sequence of homeomorphisms ψ n satisfiesthe weak category convergence condition (wcc) if:For any non-empty open set U, there is a non-empty open set V ⊆ U such that, for eachk ∈ ω,⋂V n≥k \ψ−1 n (V ) is meagre. (wcc)Equivalently, for each k ∈ ω, there is a meagre set M such that, for t /∈ M,t ∈ V =⇒ (∃n ≥ k) ψ n (t) ∈ V.For this ‘convergence to the identity’ form, see [BOst-LBII].