Normed versus topological groups: Dichotomy and duality

70 N. H. Bingham and A. J. OstaszewskiNowSoHence we obtainSo in the limitas asserted.p(x) = p(ws) ≤ p(w) + p(r) = ∑ p(z i ) + p(s)≤ ∑ bδ(1 + ε i ) + p(s)= nbδ(1 + ε) + K.p(x)‖x‖ ≤ nδ‖x‖ b(1 + ε) + M ‖x‖ .p(x)‖x‖ ≤ b(1 + ε)2 + M ‖x‖ .lim sup ‖x‖→∞p(x)‖x‖ < β,We note a related result, which requires the following definition. For p subadditive,put (for this section only)p ∗ (x) = lim inf y→x p(y),p ∗ (x) := lim sup y→x p(y).These are subadditive and lower (resp. upper) semicontinuous with p ∗ (x) ≤ p(x) ≤ p ∗ (x).Theorem 4.6 (Mueller’s Theorem – [Mue, Th. 3]). Let p be subadditive on a locallycompact group G and supposeThen p is continuous almost everywhere.lim inf x→e p ∗ (x) ≤ 0.We now return to the proof of Theorem 3.20, delayed from Section 3.2.Proof of Theorem 3.20. Apply Theorem 4.5 to the subadditive function p(x) :=‖f(x)‖, which is continuous and so Baire. Thus there is X such that, for ‖x‖ ≥ X,‖f(x)‖ ≤ β‖x‖.Taking ε = 1 in the definition of a word-net, there is δ > 0 small enough so that B δ (e) ispre-compact and there exists a compact set of generators Z δ such that for each x there isa word of length n(x) employing generators of Z δ with n(x) ≤ 2‖x‖/δ. Hence if ‖x‖ ≤ Xwe have n(x) ≤ 2M/δ. Let N := [2M/δ], the least integer greater than 2M/δ. Note thatZδN := Z δ ·...·Z δ (N times) is compact. The set B K (e) is covered by the compact swellingK :=cl[Zδ N B δ(e)]. Hence, we havesup x∈K‖f(x)‖‖x‖< ∞,(referring to β g < ∞, and continuity of ‖x‖ g /‖x‖ away from e), and soM ≤ max{β, sup x∈K ‖f(x)‖/‖x‖} < ∞.