Normed versus topological groups: Dichotomy and duality

Normed groups 97Proof. Note that‖h(tx n )h(x n ) −1 ‖ = ‖π(tx n )π(x n ) −1 ‖ = ‖π(t)‖.Hence, as π(e) = e (see Examples A4 of Section 2),and soNow{t ∈ T : h(tx n )h(x n ) −1 < n} = {t ∈ T : ‖π(t)‖ < n} = B π n(e),⋂n≥k T n(x n ) = {t ∈ T : ‖π(t)‖ < k} = B π k (e).1µ ‖t‖ X − γ ≤ ‖π(t)‖ Y ≤ 1 µ ‖t‖ X + γ,hence B π n(e) is approximated from above and below by the closed sets T ± n :T + n := {t ∈ T : 1 µ ‖t‖ X + γ ≤ n} ⊂ T (x n ) = B π n(e) ⊂ T − n := {t ∈ T : 1 µ ‖t‖ X − γ ≤ n},which yields the equivalent approximation:Hence,¯B µ(k−γ) ∩ T = {t ∈ T : ‖t‖ X ≤ µ(k − γ)} = ⋂ n≥k T + n⊂ T k (x) ⊂ ⋂ n≥k T − n = {t ∈ T : ‖t‖ X ≤ µ(k + γ)} = T ∩ ¯B µ(k+γ) .T = ⋃ k T k(x) = ⋃ k T ∩ ¯B µ(k+γ) .Hence, by the Baire Category Theorem, for some k the set T k (x) contains a Bairenon-meagre set ¯B µ(k−γ) ∩ T and the proof of Th. 7.8 applies. Indeed if T ∩ ¯B µ(k ′ +γ) isnon-meagre for some k ′ , then so is T ∩ ¯B µ(k ′ +γ) for k ≥ k ′ + 2γ and hence also T k (x) isso.Theorem 7.11 (Global bounds at infinity – Global Bounds Theorem). Let X be a locallycompact topological group with with norm having a vanishingly small global word-net.For h : X → R + , if h ∗ is globally bounded, i.e.h ∗ (u) = lim sup ‖x‖→∞ h(ux)h(x) −1 < B (u ∈ X)for some positive constant B, independent of u, then there exist constants K, L, M suchthath(ux)h(x) −1 < ‖u‖ K (u ≥ L, ‖x‖ ≥ M).Hence h is bounded away from ∞ on compact sets sufficiently far from the identity.Proof. As X is locally compact, it is a Baire space (see e.g. [Eng, Section 3.9]). Thus,by Th. 7.8, the Combinatorial Uniform Boundedness Theorem Th. 7.9A may be appliedwith T = X to a compact closed neighbourhood K = ¯B ε (e X ) of the identity e X , wherewithout loss of generality 0 < ε < 1; hence we havelim sup ‖x‖→∞ sup u∈K h(ux)h(x) −1 < ∞.