Normed versus topological groups: Dichotomy and duality

Normed groups 69Proof. By the Baire assumptions, for some k H k := {x : |f(x)| < k and |f(x −1 )| < k}is non-meagre. Note the symmetry: x ∈ H k iff x −1 ∈ H k . Suppose that f is not locallybounded; then it is not locally bounded above at some point u, i.e. there exists u n → uwithf(u n ) → +∞.Put z n := u n u −1 ; by the KBD Theorem Th. 1.2 (or Th. 5.1), for some k ∈ ω, t, t m ∈ H kand an infinite M, we haveBy symmetry, for m in M, we have{tt −1m u n u −1 t m : m ∈ M} ⊆ H k .f(u m ) = f(t m t −1 (tt −1m u m u −1 t m )t −1m u)≤ f(t m ) + f(t −1 ) + f(tt −1m u m u −1 t m ) + f(t −1m ) + f(u)≤ 4k + f(u),which contradicts f(u m ) → +∞.We recall that vanishingly small word-nets were defined in Section 3.2.Theorem 4.5. Let G be a normed group with a vanishingly small word-net. Let p : G →R + be Baire, subadditive withThenβ := lim sup ‖x‖→0+p(x)‖x‖ < ∞.lim sup ‖x‖→∞p(x)‖x‖ ≤ β < ∞.Proof. Let ε > 0. Let b = β + ε. Hence on B δ (e) for δ small enough to guarantee theexistence of Z δ and M δ we have alsop(x)‖x‖ ≤ b.By Proposition 4.4, we may assume that p is bounded by some constant K in B δ (e). Let‖x‖ > M δ .Choose a word w(x) = z 0 z 1 ...z n with ‖z i ‖ = δ(1 + ε i ) with |ε i | < ε, withp(x i ) < b‖x i ‖ = bδ(1 + ε i )andi.e.for some s with ‖s‖ < δ andd(x, w(x)) < δ,x = w(x)s1 − ε ≤ n(x)δ‖x‖ ≤ 1 + ε.