Normed versus topological groups: Dichotomy and duality

Normed groups 3Topological completeness is a natural assumption here, but it is unnecessarily strong.Respectably defined subgroups of even a compact topological group need not be G δ (see[ChMa] and [FaSol] for such examples). In Section 5 we employ the weaker notion ofalmost complete metrizability which is applicable to non-meagre Souslin-F subspaces ofa topologically complete subgroup, so embracing the non-complete examples just cited.Critical result like Th. 1.2 will be developed below using a;most completeness; elsewhere,or simplicity, we often work with topological completeness.Fresh interest in metric groups dates back to the seminal work of Milnor [Mil] in1968 on the metric properties of the fundamental group of a manifold and is key to theglobal study of manifolds initiated by Gromov [Gr1], [Gr2] in the 1980s (and we will seequasi-isometries in the duality theory of normed groups in Section 12), for which see [BH]and also [Far] for an early account; [PeSp] contains a variety of generalizations and theiruses in interpolation theory (but the context is abelian groups).The very recent [CSC] (see Sect. 2.1.1, Embedding quasi-normed groups into Banachspaces) employs norms in considering Ulam’s problem (see [Ul]) on the global approximationof nearly additive functions by additive functions. This is a topic related to regularvariation, where the weaker concept of asymptotic additivity is the key. Recall the classicaldefinition of a regularly varying function, namely a function h : R → R for which thelimit∂ R h(t) := lim x→∞ h(tx)h(x) −1(rv-limit)exists everywhere; for f Baire, the limit function is a continuous homomorphism (i.e. amultiplicative function). Following the pioneering study of [BajKar] launching a general(i.e., topological) theory of regular variation, [BOst-TRI] has re-interpreted (rv-limit), byreplacing |x| → ∞ with ‖x‖ → ∞, for functions h : X → H, with tx being the image of xunder a T -flow on X (cf. Th. 2.7 and preceding definition), and with X, T, H all groupswith right-invariant metric (right because of the division on the right) – i.e. normedgroups (making ∂h X a differential at infinity, in Michal’s sense [Michal1]). In concreteapplications the groups may be the familiar Banach groups of functional analyis, theassociated flows either the ubiquitous domain translations of Fourier transform theoryor convolutions from the related contexts of abstract harmonic analysis (e.g. Wiener’sTauberian theory so relevant to classical regular variation – see e.g. [BGT, Ch. 4]). In allof these one is guaranteed right-invariant metrics. Likewise in the foundations of regularvariation the first tool is the group H(X) of bounded self-homeomorphisms of the groupX under a supremum metric (and acting transitively on X); the metric is again rightinvariantand hence a group-norm. It is thus natural, in view of the applications and theBirkhoff-Kakutani Theorem, to favour right-invariance.We show in Section 4 and 10 that normed groups offer a natural setting for subadditivityand for (mid-point) convexity.