Normed versus topological groups: Dichotomy and duality

124 N. H. Bingham and A. J. OstaszewskiConsider x with ‖x‖ < δ N (η). Then κ N ‖f(x)‖ ≤ ‖f(x) N ‖ = ‖f(x N )‖ < M. So for xwith ‖x‖ < δ N (η) we have‖f(x)‖ < M/κ N < ε,proving continuity at e.Compare [HJ, Th 2.4.9 p. 382]. The Main Theorem of [BOst-Thin] may be given acombinatorial restatement in the group setting. We need some further definitions.Definition. For G a metric group, let C(G) = C(N, G) := {x ∈ G N : x is convergent}denote the sequence space of G. For x ∈ C(G) we writeWe make C(G) into a group by settingL(x) = lim n x n .x · y : = 〈x n y n : n ∈ N〉.Thus e = 〈e G 〉 and x −1 = 〈x −1n 〉. We identify G with the subgroup of constant sequences,that isT = {〈g : n ∈ N〉 : g ∈ G}.The natural action of G or T on C(G) is then tx := 〈tx n : n ∈ N〉. Thus 〈g〉 = ge, andthen tx = te · x.Definition. For G a group, a set G of convergent sequences u = 〈u n : n ∈ N〉 in c(G) isa G-ideal in the sequence space C(G) if it is a subgroup closed under the mutiplicativeaction of G, and will be termed complete if it is closed under subsequence formation.That is, a complete G-ideal in C(G) satisfies(i) u ∈ G implies tu = 〈tu n 〉 ∈ G, for each t in G,(ii) u, v ∈ G implies that uv −1 ∈ G,(iii) u ∈ G implies that u M := {u m : m ∈ M} ∈ G for every infinite M.If G satisfies (i) and u, v ∈ G implies only that uv ∈ G, we say that G is a G-subidealin C(G).Remarks. 0. In the notation of (iii) above, if G is merely an ideal then G ∗ = {u M : foru ∈ t and M ⊂ N} is a complete G-ideal; indeed tu M = (tu) Mand u M v −1M= (uv−1 ) Mand u MM ′ = u M ′ for M ′ ⊂ M.1. We speak of a Euclidean sequential structure when G is the vector space R d regardedas an additive group.2. The conditions (i) and (ii) assert that G is similar in structure to a left-ideal, beingclosed under multiplication by G and a subgroup of C(G).3. We refer only to the combinatorial properties of C(G); but one may give C(G) a pseudonormby setting‖x‖ c := d G (Lx, e) = ‖Lx‖, where Lx := lim x n .