54 N. H. Bingham <strong>and</strong> A. J. OstaszewskiTheorem 3.37. For X a normed group, if products of Cauchy sequences are Cauchythen X is a <strong>topological</strong> group.Proof. By Th.3.35 (C-adm) holds. The latter implies the weak admissibility conditionwhich, by Lemma 3.33 <strong>and</strong> Th. 3.4, implies that X is a <strong>topological</strong> group.Definitions. 1. Say that a normed group is bi-Cauchy complete if each bi-Cauchy sequencehas a limit.2. For any metric d on X, the relation {x n } ∼ d {y n } between d-Cauchy sequences, definedby requiring d(x n , y n ) → 0 (as n → ∞), is an equivalence (by the triangle inequality).In particular for d S = max{d R , d L }, it defines an equivalence {x n } ∼ {y n } betweenbi-Cauchy sequences. It is the intersection of the right <strong>and</strong> left equivalence relations, dem<strong>and</strong>ingboth d R (x n , y n ) → 0 <strong>and</strong> d L (x n , y n ) → 0.3. For X a normed group, working modulo ∼ putWorking in ˜X define˜X := {{x n } : {x n } is a bi-Cauchy sequence}.‖{x n }‖ : = lim n ‖x n ‖, <strong>and</strong> {x n } · {y n } = lim n x n y n ,˜d R ({x n }, {y n }) : = lim n ‖x n y −1n‖ <strong>and</strong> ˜d L ({x n }, {y n }) := lim n ‖x −1n y n ‖.(Compare also the sequence space C(G) considered in Section 11.)The following result is in a thin disguise the st<strong>and</strong>ard result on the completion of a<strong>topological</strong> group under its ambidextrous uniformity (as under our assumptions X is infact a <strong>topological</strong> space), see e.g. [Kel, Problem 6Q]; here we are merely asserting additionallythat the completion uniformity extends the originating norm <strong>and</strong> is normable,provided the uniformity on X is.Of course ˜X need not be ˜d R -complete; indeed it will not be if there are Cauchy sequencesthat are not bi-Cauchy. However, ˜X under ˜dR is <strong>topological</strong>ly complete. Indeedwe have ˜d S = max{ ˜d R , ˜d L }, <strong>and</strong> by construction ( ˜X, ˜d S ) is complete, <strong>and</strong> being a <strong>topological</strong>group is homeomorphic to ( ˜X, ˜d R ), by the Ambidextrous Refinement Principle(Th. 3.9). Note that ( ˜X, ˜d R ) as a metric space has a completion ( ˆX, ˆd), not necessarily agroup, in which of course (˜X, ˜d R ) is embedded as a G δ set.Theorem 3.38 (Bi-Cauchy completion). If the group-norm of X satisfies (C-adm), then˜X is a normed group extending X (isometrically), satisfying (C-adm) (so also a <strong>topological</strong>group), in which bi-Cauchy sequences are convergent.Proof. We work under the right norm topology. By Th.3.35, products of Cauchy sequencesin X are Cauchy. Note that {x n } · {y n } ∼ {e} implies that {y n } ∼ {x −1n }. So X ∼ is
<strong>Normed</strong> <strong>groups</strong> 55equipped with an inversion.We now verify that ‖ · ‖ is indeed a norm on X ∼ . We have<strong>and</strong>so {x n } ∼ {e}; also‖{x n } · {y n }‖ = lim n ‖x n y n ‖ ≤ lim n ‖x n ‖ + lim n ‖y n ‖,‖{x n }‖ = 0 iff lim n ‖x n ‖ = 0 iff x n → e,‖{x −1nWe note that if x n → R x, then}‖ = lim n ‖x −1n ‖ = lim n ‖x n ‖ = ‖{x n }‖.‖{x n } · {x −1 }‖ = lim n ‖x n x −1 ‖ = 0,so that {x n } → R {x} <strong>and</strong> hence the map x → {x n } where x n ≡ x isometrically embedsX into X ∼ . This far X ∼ is a normed group. Say that {x n } is d-regular if d(x n , x m ) ≤ 2 −nfor m ≥ n. If {{x m n } n } m is ˜d R -regular with each {x m n } also d R -regular, put y n = x n n. Then{y n } is the limit of {{x m n } n } m .Notice that if {w m n } → e, then without loss of generality w n n is null, so we have<strong>and</strong> so X ∼ also satisfies (C-adm).x n nw n n(x n n) −1 → e X ,Remarks. 1. The definition of ˜X requires sequences to be bi-Cauchy to achieve bi-Cauchy completeness. Compare this two-sided condition to that of Prop. 3.13 whichuses bi-uniformly continuous functions, <strong>and</strong> also [BePe] Prop 1.1, where in the context ofAuth(X) with the weak refinement topology (that defined in Th. 2.12, as opposed to thatof Th. 3.19, where there is an abelian norm), the two-sided assumptions lim n f n = f ∈ X X<strong>and</strong> lim n f −1 = g ∈ X X (limits in the supremum metric) yield g = f −1 ∈ Auth(X). (Onthis last point see also Lemma 1 of [Ost-Joint].)2. If X is complete under d R there is no guarantee that X is closed under products ofCauchy sequences, so Th. 3.35 does not characterize (C-adm).We now consider the impact of automatic continuity. Our first result captures theeffect on automorphisms of the result, due to Darboux [Dar], that an additive functionwhich is locally bounded is continuous.Definition. Say that a group is Darboux-normed if there are constants κ n with κ n → ∞associated with the group-norm such that for all elements z of the groupor equivalentlyκ n ‖z‖ ≤ ‖z n ‖,‖z 1/n ‖ ≤ 1κ n‖z‖.Thus z 1/n → e; a related condition was considered by McShane in [McSh] (cf. theEberlein-McShane Theorem, Th. 10.1).
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N. H. BINGHAM and A. J. OSTASZEWSKI
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Normed groups 3ContentsContents . .
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1. IntroductionGroup-norms, which b
- Page 7 and 8: Normed groups 3Topological complete
- Page 9 and 10: Normed groups 5abelian group has se
- Page 11 and 12: Normed groups 74 (Topological permu
- Page 13 and 14: Normed groups 9The following result
- Page 15 and 16: Normed groups 11Corollary 2.4. For
- Page 17 and 18: Normed groups 13More generally, for
- Page 19 and 20: Normed groups 15definitions, our pr
- Page 21 and 22: Normed groups 17so that fg is in th
- Page 23 and 24: Normed groups 19(iii) The ¯d H -to
- Page 25 and 26: Normed groups 21so‖αβ‖ ≤
- Page 27 and 28: Normed groups 23Remark. Note that,
- Page 29 and 30: Normed groups 25shows that [z n , y
- Page 31 and 32: Normed groups 27Denoting this commo
- Page 33 and 34: Normed groups 29Theorem 3.4 (Equiva
- Page 35 and 36: Normed groups 31argument as again p
- Page 37 and 38: Normed groups 33(ii) For α ∈ H u
- Page 39 and 40: Normed groups 35Definition. A group
- Page 41 and 42: Normed groups 37We now give an expl
- Page 43 and 44: Normed groups 39Theorem 3.19 (Abeli
- Page 45 and 46: Normed groups 412. Further recall t
- Page 47 and 48: Normed groups 43Theorem 3.22 (Lipsc
- Page 49 and 50: Normed groups 45Proof. Z γ = G (cf
- Page 51 and 52: Normed groups 47Theorem 3.30. Let G
- Page 53 and 54: Normed groups 49Remark. On the matt
- Page 55 and 56: Normed groups 51As for the conclusi
- Page 57: Normed groups 53By (C-adm), we may
- Page 61 and 62: Normed groups 57Proof. To apply Th.
- Page 63 and 64: Normed groups 59Definition. A point
- Page 65 and 66: Normed groups 61Proposition 3.46 (M
- Page 67 and 68: Normed groups 63Thus ω δ (s) ≤
- Page 69 and 70: Normed groups 65Remark. In the penu
- Page 71 and 72: Normed groups 67The result confirms
- Page 73 and 74: Normed groups 69Proof. By the Baire
- Page 75 and 76: Normed groups 715. Generic Dichotom
- Page 77 and 78: Normed groups 73Returning to the cr
- Page 79 and 80: Normed groups 75Examples. Here are
- Page 81 and 82: Normed groups 77cf. [Eng, 4.3.23].)
- Page 83 and 84: Normed groups 79Remarks. 1. See [Fo
- Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
- Page 87 and 88: Normed groups 83is continuous at th
- Page 89 and 90: Normed groups 85compact. Evidently,
- Page 91 and 92: Normed groups 87j ∈ ω} which enu
- Page 93 and 94: Normed groups 89The result below ge
- Page 95 and 96: Normed groups 91left-shift, not in
- Page 97 and 98: Normed groups 93As a corollary of t
- Page 99 and 100: Normed groups 953. For X a normed g
- Page 101 and 102: Normed groups 97Proof. Note that‖
- Page 103 and 104: Normed groups 99Taking h(x) := ‖
- Page 105 and 106: Normed groups 1019. The Semigroup T
- Page 107 and 108: Normed groups 103Theorem 9.5 (Semig
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Normed groups 105By the Category Em
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Normed groups 107Proof. Say f is bo
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Normed groups 109Thus G is locally
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Normed groups 111Theorem 10.10 (Bar
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Normed groups 113K-analyticity was
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Normed groups 115Theorem 11.6 (Disc
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Normed groups 117restricted to X\M
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Normed groups 119groups need not be
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Normed groups 121Proof. In the meas
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Normed groups 123Hence, as t i n
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Normed groups 125The corresponding
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Normed groups 127(t, x) ✛✻Φ T
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Normed groups 129Fix s. Since s is
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Normed groups 131Hence,‖x‖ −
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Normed groups 133converging to the
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Normed groups 135Definition. Let {
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Normed groups 137However, whilst th
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Normed groups 139embeddable, 14enab
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Normed groups 141Bibliography[AL]J.
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Normed groups 143Series 378, 2010.[
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Normed groups 145abelian groups, Ma
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Normed groups 147[Kak] S. Kakutani,
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Normed groups 149fields. I. Basic p
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Normed groups 151[So]R. M. Solovay,