Normed versus topological groups: Dichotomy and duality

Normed groups 67The result confirms that if λ t is quasi-continuus then it is subcontinuous on a denseset of points, a fortiori at one point, and so at e by the remarks to the definition ofsubcontinuity.4. SubadditivityDefinition. Let X be a normed group. A function p : X → R is subadditive ifp(xy) ≤ p(x) + p(y).Thus a norm ‖x‖ and so also any g-conjugate norm ‖x‖ g are examples. Recall from [Kucz,p.140] the definitions of upper and lower hulls of a function p :M p (x) = lim r→0+ sup{p(z) : z ∈ B r (x)},m p (x) = lim r→0+ inf{p(z) : z ∈ B r (x)}.(Usually these are of interest for convex functions p.) These definitions remain valid fora normed group. (Note that e.g. inf{p(z) : z ∈ B r (x)} is a decreasing function of r.) Weunderstand the balls here to be defined by a right-invariant metric, i.e.B r (x) := {y : d(x, y) < r} with d right-invariant.These are subadditive functions if the group G is R d . We reprove some results fromKuczma [Kucz], thus verifying the extent to which they may be generalized to normedgroups. Only our first result appears to need the Klee property (bi-invariance of themetric); fortunately this result is not needed in the sequel. The Main Theorem belowconcerns the behaviour of p(x)/‖x‖.Lemma 4.1 (cf. [Kucz, L. 1 p. 403]).and M p are subadditive.For a normed group G with the Klee property, m pProof. For a > m p (x) and b > m p (y) and r > 0, let d(u, x) < r and d(v, y) < r satisfyinf{p(z) : z ∈ B r (x)} ≤ p(u) < a, and inf{p(z) : z ∈ B r (y)} ≤ p(v) < b.Then, by the Klee property,d(xy, uv) ≤ d(x, u) + d(y, v) < 2r.Nowinf{p(z) : z ∈ B 2r (xy)} ≤ p(uv) ≤ p(u) + p(v) < a + b,henceinf{p(z) : z ∈ B 2r (xy)} ≤ inf{p(z) : z ∈ B r (x)} + inf{p(z) : z ∈ B r (x)},and the result follows on taking limits as r → 0 + .