Normed versus topological groups: Dichotomy and duality

26 N. H. Bingham and A. J. Ostaszewskishowing that the product of two slowly varying functions is slowly varying, sincef(tx)f(t) −1 → e iff ‖f(tx)f(t) −1 ‖ → 0.3. Normed versus topological groupsBy the Birkhoff-Kakutani Theorem above (Th. 2.19) any metrizable topological grouphas a right-invariant equivalent metric, and hence is a normed group. Theorem 3.4 belowestablishes a converse: a normed group is a topological group provided all its shifts(both right and left-sided) are continuous, i.e. provided the normed group is semitopological(see [ArRez]). This is not altogether surprising, in the light of known results onsemitopological groups: assuming that a group T is metrizable, non-meagre and analyticin the metric, and that both left and right-shifts are continuous, then T is a topologicalgroup (see e.g. [THJ] for several results of this kind in [Rog2, p. 352]; compare also[Ell2] and the literature cited under Remarks 2 in Section 2). The results here are cognate,and new because a normed group has a one-sided rather than a two-sided topology.We will also establish the equivalent condition that all conjugacies γ g (x) := gxg −1 arecontinuous; this has the advantage of being stated in terms of the norm, rather than interms of one of the associated metrics. As inner automorphisms are homomorphisms, thiscondition ties the structure of normed groups to issues of automatic continuity of homomorphisms:automatic continuity forces a normed group to be a topological group (andthe homomorphisms to be homeomorphisms). Normed groups are thus either topologicalor pathological, as noted in the Introduction.The current section falls into three parts. In the first we characterize topologicalgroups in the category of normed groups and so in particular, using norms, characterizealso the Klee groups (topological groups which have an equivalent bi-invariant metric).Then we study continuous automorphisms in relation to Lipschitz norms. In the thirdsubsection we demonstrate that a small amount of regularity forces a normed group tobe a topological group.3.1. Left versus right-shifts: Equivalence Theorem. As we have seen in Th. 2.3, agroup-norm defines two metrics: the right-invariant metric which we denote as usual byd R (x, y) := ‖xy −1 ‖ and the conjugate left-invariant metric, here to be denoted d L (x, y) :=d R (x −1 , y −1 ) = ‖x −1 y‖. There is correspondingly a right and left metric topology whichwe term the right or left norm topology. We favour this over ‘right’ or ‘left’ normed groupsrather than follow the [HS] paradigm of ‘right’ and ‘left’ topological semigroups. We write→ R for convergence under d R etc. Recall that both metrics give rise to the same norm,since d L (x, e) = d R (x −1 , e) = d R (e, x) = ‖x‖, and hence define the same balls centeredat the origin e:B d R(e, r) := {x : d(e, x) < r} = B d L(e, r).