122 N. H. Bingham <strong>and</strong> A. J. OstaszewskiEnumerate the countable family {H(α|n) : α ∈ ω n } as {T h : h ∈ ω}. Since S spans G,we haveG = ⋃ ⋃(Th∈ω k∈N h k 1· ... · T kh ) .As each T k is analytic, so too is the continuous imageT k1 · ... · T kh ,which is thus measurable. Hence, for some h ∈ N <strong>and</strong> k ∈ N h the setT k1 · ... · T khhas positive measure/ is non-meagre.Definition. We say that S is a pre-compact set if its closure is compact. We will saythat f is a pre-compact function if f(S) is pre-compact for each pre-compact set S.Theorem 11.21 (Jones-Kominek Analytic Automaticity Theorem for Metric Groups).Let G be either a non-meagre normed <strong>topological</strong> group, or a <strong>topological</strong> group supportinga Radon measure, <strong>and</strong> let H be K-analytic (hence Lindelöf, <strong>and</strong> so second countable inour metric setting). Let h : G → H be a homomorphism between metric <strong>groups</strong> <strong>and</strong> let Tbe an analytic set in G which finitely generates G.(i) (Jones condition) If h is continuous on T, then h is continuous.(ii) (Kominek condition) If h is pre-compact on T, then h is precompact.Proof. As in the Analytic Covering Lemma (Th. 11.19), writeT = ⋃ k∈N T k.(i) If h is not continuous, suppose that x n → x 0 but h(x n ) does not converge to h(x 0 ).SinceG = ⋃ ⋃T (m)m∈N k∈Nk,G is a union of analytic sets <strong>and</strong> hence analytic ([Jay-Rog, Th. 2.5.4 p. 23]). Now, forsome m, k the m-span T (m)kis non-meagre, as is the m-span S (m)kof some compact subsetS k ⊆ T k . So for some shifted subsequence tx n → tx 0 , where t <strong>and</strong> x 0 lie in S (m)k. Thusthere is an infinite set M such that, for n ∈ M,tx n = t 1 n...t m n with t i n ∈ S k .Without loss of generality, as S k is compact,<strong>and</strong> sot (i)n→ t (i)0 ∈ S k ⊂ T,tx n = t 1 n...t m n → t 1 0...t m 0 = tx 0 with t i 0 ∈ S k ⊂ T.
<strong>Normed</strong> <strong>groups</strong> 123Hence, as t i n → t i 0 ⊂ T, we have, for n ∈ M,Thush(t)h(x n ) = h(tx n ) = h(t 1 n...t m n ) = h(t 1 n)...h(t m n )→ h(t 1 0)...h(t m 0 ) = h(t 1 0...t m 0 )= h(tx 0 ) = h(t)h(x 0 ).h(x n ) → h(x 0 ),a contradiction.(ii) If {h(x n )} is not precompact with {x n } precompact, by the same argument, for someS (n)k<strong>and</strong> some infinite set M, we have tx n = t 1 n...t m n <strong>and</strong> t i n → t i 0 ⊂ T , for n ∈ M. Henceh(tx n ) = h(t)h(x n ) is precompact <strong>and</strong> so h(x n ) is precompact, a contradiction.The following result connects the preceeding theorem to Darboux’s Theorem, that alocally bounded additive function on the reals is continuous ([Dar], or [AD]).Definition. Say that a homomorphism between normed <strong>groups</strong> is N-homogeneous if‖f(x n )‖ = n‖f(x)‖, for any x <strong>and</strong> n ∈ N (cf. Section 2 where N-homogeneous normswere considered, for which homomorphisms are automatically N-homogeneous). Thusany homomorphism into the additive reals is N-homogeneous. Recall from Section 3.3that the norm is a Darboux norm (or, in this context N-subhomogeneous) if there areconstants κ n with κ n → ∞ 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). In keeping with the convention of functionalanalysis (appropriately to our usage of norm) the next result refers to a locally boundedhomomorphism as bounded.Theorem 11.22 (Generalized Darboux Theorem – [Dar]). A bounded homomorphismfrom a normed group to a Darboux normed group (N-subhomogeneous norm) is continuous.In particular, a bounded, additive function on R is continuous.Proof. Suppose that f : G → H is a homomorphism to a normed N-subhomogeneousgroup H; thus ‖f(x n )‖ ≥ κ n ‖f(x)‖, for any x ∈ G <strong>and</strong> n ∈ N. Suppose that f is boundedby M <strong>and</strong>, for ‖x‖ < η, we have‖f(x)‖ < M.Let ε > 0 be given. Choose N such that κ N > M/ε, i.e. M/κ N < ε. Now x → x N iscontinuous, hence there is δ = δ N (η) > 0 such that, for ‖x‖ < δ,‖x N ‖ < η.
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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
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Normed groups 3Topological complete
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Normed groups 5abelian group has se
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Normed groups 74 (Topological permu
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Normed groups 9The following result
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Normed groups 11Corollary 2.4. For
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Normed groups 13More generally, for
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Normed groups 15definitions, our pr
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Normed groups 17so that fg is in th
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Normed groups 19(iii) The ¯d H -to
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Normed groups 21so‖αβ‖ ≤
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Normed groups 23Remark. Note that,
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Normed groups 25shows that [z n , y
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Normed groups 27Denoting this commo
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Normed groups 29Theorem 3.4 (Equiva
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Normed groups 31argument as again p
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Normed groups 33(ii) For α ∈ H u
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Normed groups 35Definition. A group
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Normed groups 37We now give an expl
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Normed groups 39Theorem 3.19 (Abeli
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Normed groups 412. Further recall t
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Normed groups 43Theorem 3.22 (Lipsc
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Normed groups 45Proof. Z γ = G (cf
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Normed groups 47Theorem 3.30. Let G
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Normed groups 49Remark. On the matt
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Normed groups 51As for the conclusi
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Normed groups 53By (C-adm), we may
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Normed groups 55equipped with an in
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Normed groups 57Proof. To apply Th.
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Normed groups 59Definition. A point
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Normed groups 61Proposition 3.46 (M
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Normed groups 63Thus ω δ (s) ≤
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Normed groups 65Remark. In the penu
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Normed groups 67The result confirms
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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
- Page 109 and 110: Normed groups 105By the Category Em
- Page 111 and 112: Normed groups 107Proof. Say f is bo
- Page 113 and 114: Normed groups 109Thus G is locally
- Page 115 and 116: Normed groups 111Theorem 10.10 (Bar
- Page 117 and 118: Normed groups 113K-analyticity was
- Page 119 and 120: Normed groups 115Theorem 11.6 (Disc
- Page 121 and 122: Normed groups 117restricted to X\M
- Page 123 and 124: Normed groups 119groups need not be
- Page 125: Normed groups 121Proof. In the meas
- Page 129 and 130: Normed groups 125The corresponding
- Page 131 and 132: Normed groups 127(t, x) ✛✻Φ T
- Page 133 and 134: Normed groups 129Fix s. Since s is
- Page 135 and 136: Normed groups 131Hence,‖x‖ −
- Page 137 and 138: Normed groups 133converging to the
- Page 139 and 140: Normed groups 135Definition. Let {
- Page 141 and 142: Normed groups 137However, whilst th
- Page 143 and 144: Normed groups 139embeddable, 14enab
- Page 145 and 146: Normed groups 141Bibliography[AL]J.
- Page 147 and 148: Normed groups 143Series 378, 2010.[
- Page 149 and 150: Normed groups 145abelian groups, Ma
- Page 151 and 152: Normed groups 147[Kak] S. Kakutani,
- Page 153 and 154: Normed groups 149fields. I. Basic p
- Page 155: Normed groups 151[So]R. M. Solovay,