Normed versus topological groups: Dichotomy and duality

104 N. H. Bingham and A. J. Ostaszewskiwhere the equations over G are soluble (compare also [Lyn1]).2. In an arbitrary group, say that a subset C is 1 2-convex if, for all x, yx, y ∈ C =⇒ √ xy ∈ C,where √ xy signifies some element z with z 2 = xy. We recall the following results.Theorem 10.1 (Eberlein-McShane Theorem, [Eb], [McSh, Cor. 10]). Let X be a 2-divisible topological group of second category. Then any 1 2-convex non-meagre Baire sethas a non-empty interior. If X is abelian and each sequence defined by x 2 n+1 = x n convergesto e X then the interior of a 1 2-convex set C is dense in C.Definition. We say that the function h : G → R is 1 2 -convex on the 1 2–convex set C if,for x, y ∈ C,with √ xy as above.h( √ xy) ≤ 1 (h(x) + h(y)) ,2Example. For G = R+ ∗ the function h(x) = x is 1 2-convex on G, since2xy ≤ x 2 + y 2 .Theorem 10.2 (Convex Minorant Theorem, [McSh]). Let X be 2-divisible abelian topologicalgroup. Let f and g be real-valued functions defined on a non-meagre subset C withf 1 2 -convex and g Baire such that f(x) ≤ g(x), for x ∈ C.Then f is continuous on the interior of C.Lemma 10.3 (Averaging Lemma). In a normed topological group, a non-meagre set T is‘averaging’, that is, for any given point u ∈ T and for any sequence {u n } → u, there arev ∈ G (a right-averaging translator) and {v n } ⊆ T such that, for infinitely many n ∈ ω,we haveu 2 n = v n v.There is likewise a left-averaging translator such that for some {w n } ⊆ T for infinitelymany n ∈ ω, we haveu 2 n = ww n .Proof. Define null sequences byz n = u n u −1 , and ˜z n = u −1 u n . We are to solve u 2 nv −1 =v n ∈ T, oru˜z n z n uv −1 = v n ∈ T, equivalently ˜z n z n uv −1 = u −1 v n ∈ T ′ = u −1 T.Now put ψ n (x) := ˜z n z n x; thend(x, ˜z n z n x) = d(e, ˜z n z n ) = ‖˜z n z n ‖ ≤ ‖˜z n ‖ + ‖z n ‖ → 0.