Normed versus topological groups: Dichotomy and duality

Normed groups 137However, whilst this is not a countably generated filter, its projection on the x-coordinate:{α : d X (α n (x), x) > M ultimately},is.4. When the group is locally compact, ‘bounded’ may be defined as ‘pre-compact’, and so‘divergent’ becomes ‘unbounded’. Here divergence is unconditional (because continuitypreserves compactness).Theorem 13.6. For A ⊆ H(S), pointwise divergence in A is unconditional.Proof. For fixed s ∈ S, σ ∈ H(S) and d X (α n (s), s)) unbounded, suppose thatd X (σα n (s), s)) is bounded by K. Thend S (α n (s), s)) ≤ d S (α n (s), σ(α n (s))) + d S (σ(α n (s)), s)≤ ‖σ‖ H(S) + K,contradicting that d S (α n (s), s)) is unbounded. Similarly, for ψ n converging to the identity,if d S (ψ n (α n (x)), x) is bounded by L, thend S (α n (s), s)) ≤ d S (α n (s), ψ n (α n (s))) + d S (ψ n (α n (s)), s)≤ ‖ψ n ‖ H(S) + L,contradicting that d S (α n (s), s)) is unbounded.Corollary 13.7. Pointwise divergence in A ⊆ H(X) is unconditional.Corollary 13.8. Pointwise divergence in A = Ξ is unconditional.Proof. In Theorem 13.6, take α n = ξ x(n) . Then unboundedness of d T (ξ x(n) (t), t) impliesunboundedness of d T (σξ x(n) (t), t) and of d T (ψ n ξ x(n) (t)), t).Acknowledgement. We are very grateful to the referee for his wise, scholarly andextensive comments which have both greatly improved the exposition of this paper andled to some new results. We also thank Roman Pol for his helpful comments and drawingour attention to relevant literature.