Normed versus topological groups: Dichotomy and duality

Normed groups 23Remark. Note that, taking b = v = e, we have the triangle inequality. Thus the result(a) characterizes maps ‖ · ‖ with the positivity property as group pre-norms which areabelian. In regard to conjugacy, see also the Uniformity Theorem for Conjugation (Th.12.4). We now state the following classical result.Theorem 2.19 (Normability Theorem for Groups – Birkhoff-Kakutani Theorem). LetX be a first-countable topological group and let V n be a symmetric local base at e X withV 4 n+1 ⊆ V n .Let r = ∑ ∞n=1 c n(r)2 −n be a terminating representation of the dyadic number r, and putA(r) := ∑ ∞n=1 c n(r)V n .Thenp(x) := inf{r : x ∈ A(r)}is a group-norm. If further X is locally compact and non-compact, then p may be arrangedsuch that p is unbounded on X, but bounded on compact sets.For a proof see that offered in [Ru] for Th. 1.24 (p. 18-19), which derives a metrizationof a topological vector space in the form d(x, y) = p(x − y) and makes no use of thescalar field (so note how symmetric neighbourhoods here replace the ‘balanced’ ones in atopological vector space). That proof may be rewritten verbatim with xy −1 substitutingfor the additive notation x − y (cf. Proposition 2.2).Remark. In fact, a close inspection of Kakutani’s metrizability proof in [Kak] (cf. [SeKu]§7.4) for topological groups yields the following characterization of normed groups – fordetails see [Ost-LB3].Theorem 2.19 ′ (Normability Theorem for right topological groups – Birkhoff-KakutaniTheorem). A first-countable right topological group X is a normed group iff inversionand multiplication are continuous at the identity.We close with some information concerning commutators, which arise in Theorems3.7, 6.3, 10.7 and 10.9.Definition. The right-sided and left-sided commutators are defined byAs[x, y] L : = xyx −1 y −1 .[x, y] R : = x −1 y −1 xy = [x −1 , y −1 ] L .xy = [x, y] L yx and xy = yx[x, y] R ,these express in terms of shifts the distortion arising from commuting factors, and sotheir continuity here is significant. Let [x, y] denote either a right or left commutator;