Normed versus topological groups: Dichotomy and duality

126 N. H. Bingham and A. J. OstaszewskiThis theorem is applied with G = R d and H = R in [BOst-Aeq] to subadditivefunctions, convex functions, and to regularly varying functions (defined on R d ) to deriveautomatic properties such as automatic continuity, automatic local boundedness andautomatic uniform boundedness.12. Duality in normed groupsIn this section – to distinguish two contexts – we use the generic notation of S for a groupwith metric d S ; recall from Section 3 that Auth(S) denotes the self-homeomorphisms(auto-homeomorphisms) of S; H(S) denotes the bounded elements of Auth(S). We writeA ⊆ H(S) for a subgroup of self-homeomorphisms of S. We work in the category ofnormed groups. However, by specializing to A = H u (S), the homeomorphisms that arebi-uniformly continuous (relative to d S ), we can regard the development as also takingplace inside the category of topological groups, by Th. 3.13. We assume that A is metrizedby the supremum metricd A (t 1 , t 2 ) = sup s∈S d S (t 1 (s), t 2 (s)).Note that e A = id S . The purpose of this notation is to embrace the two cases: (i) S = Xand A = H u (X), and (ii) S = H u (X) and A = H u (H u(X)). In what follows, we regardthe group H u (X) as the topological (uniform) dual of X and verify that (X, d X ) is embeddedin the second dual H u (H u (X)). As an application one may use this duality toclarify, in the context of a non-autonomous differential equation with initial conditions,the link between its solutions trajectories and flows of its varying ‘coefficient matrix’.See [Se1] and [Se2], which derive the close relationship for a general non-autonomousdifferential equation u ′ = f(u, t) with u(0) = x ∈ X, between its trajectories in X andlocal flows in the function space Φ of translates f t of f (where f t (x, s) = f(x, t + s)).One may alternatively capture the topological duality as algebraic complementarity – see[Ost-knit] for details. A summary will suffice here. One first considers the commutativediagram below where initially the maps are only homeomorphisms (herein T ⊆ H u (X)and Φ T (t, x) = (t, tx) and Φ X (x, t) = (t, xt) are embeddings). Then one extends the diagramto a diagram of isomorphisms, a change facilitated by forming the direct productgroup G := T × X. Thus G = T G X G where T G and X G are normal subgroups, commutingelementwise, and isomorphic respectively to T and X; moreover, the subgroupT G , acting multiplicatively on X G , represents the T -flow on X and simultaneously themultiplicative action of X G on G represents the X-flow on T X = {t x : t ∈ T, x ∈ X},the group of right-translates of T , where t x (u) = θ x (t)(u) = t(ux). If G has an invariantmetric d G , and T G and X G are now regarded as groups of translations on G, then theymay be metrized by the supremum metric ˆd G , whereupon each is isometric to itself assubgroup of G. Our approach here suffers a loss of elegance, by dispensing with G, butgains analytically by working directly with d X and ˆd X .