68 N. H. Bingham <strong>and</strong> A. J. OstaszewskiLemma 4.2 (cf. [Kucz, L. 2 p. 403]). For a normed group G, if p : G → R is subadditive,thenm p (x) ≤ M p (x) <strong>and</strong> M p (x) − m p (x) ≤ M p (e).Proof. Only the second assertion needs proof. For a > m p (x) <strong>and</strong> b < M p (x), there existu, v ∈ B r (x) witha > p(u) ≥ m p (x), <strong>and</strong> b < p(v) ≤ M p (x).SoNowb − a < p(v) − p(u) ≤ p(vu −1 u) − p(u) ≤ p(vu −1 ) + p(u) − p(u) = p(vu −1 ).so vu −1 ∈ B 2r (e), <strong>and</strong> hence‖vu −1 ‖ ≤ ‖v‖ + ‖u‖ < 2r,p(vu −1 ) ≤ sup{p(z) : z ∈ B 2r (e)}.Hence, with r fixed, taking a, b to their respective limits,M p (x) − m p (x) ≤ sup{p(z) : z ∈ B 2r (e)}.Taking limits as r → 0+, we obtain the second inequality.Lemma 4.3. For a normed group G <strong>and</strong> any subadditive function f : G → R, if f islocally bounded above at a point, then it is locally bounded at every point.Proof. We repeat the proof in [Kucz, Th. 2, p.404], thus verifying that it continues tohold in a normed group.Suppose that p is locally bounded above at t 0 by K. We first show that f is locallybounded above at e. Suppose otherwise that for some t n → e we have p(t n ) → ∞. Nowt n t 0 → et 0 = t 0 , <strong>and</strong> sop(t n ) = p(t n t 0 t −10 ) ≤ p(t nt 0 ) + p(t −10 ) ≤ K + p(t−1 0 ),a contradiction. Hence p is locally bounded above at e, i.e. M p (e) < ∞. But 0 ≤ M p (x) −m p (x) ≤ M p (e), hence both M p (x) <strong>and</strong> m p (x) are finite for every x. That is, p is locallybounded above <strong>and</strong> below at each x.The next result requires that both f(x) <strong>and</strong> f(x −1 ) be Baire functions; this happensfor instance when (i) f is even, i.e. f(x) = f(x −1 ), with f(x) := ‖gxg −1 ‖ an example ofsome interest here (cf. Th. 3.27 <strong>and</strong> in connection with the oscillation function of Section3.3), <strong>and</strong> (ii) both f(x) <strong>and</strong> x → x −1 are Baire, so that the normed group is a <strong>topological</strong>group (Th. 3.41).Proposition 4.4 ([Kucz, Th. 3, p. 404]). For a <strong>topological</strong>ly complete normed group G<strong>and</strong> a Baire function f : G → R with x → f(x −1 ) Baire, if f is subadditive, then f islocally bounded.
<strong>Normed</strong> <strong>groups</strong> 69Proof. By the Baire assumptions, for some k H k := {x : |f(x)| < k <strong>and</strong> |f(x −1 )| < k}is non-meagre. Note the symmetry: x ∈ H k iff x −1 ∈ H k . Suppose that f is not locallybounded; then it is not locally bounded above at some point u, i.e. there exists u n → uwithf(u n ) → +∞.Put z n := u n u −1 ; by the KBD Theorem Th. 1.2 (or Th. 5.1), for some k ∈ ω, t, t m ∈ H k<strong>and</strong> an infinite M, we haveBy symmetry, for m in M, we have{tt −1m u n u −1 t m : m ∈ M} ⊆ H k .f(u m ) = f(t m t −1 (tt −1m u m u −1 t m )t −1m u)≤ f(t m ) + f(t −1 ) + f(tt −1m u m u −1 t m ) + f(t −1m ) + f(u)≤ 4k + f(u),which contradicts f(u m ) → +∞.We recall that vanishingly small word-nets were defined in Section 3.2.Theorem 4.5. Let G be a normed group with a vanishingly small word-net. Let p : G →R + be Baire, subadditive withThenβ := lim sup ‖x‖→0+p(x)‖x‖ < ∞.lim sup ‖x‖→∞p(x)‖x‖ ≤ β < ∞.Proof. Let ε > 0. Let b = β + ε. Hence on B δ (e) for δ small enough to guarantee theexistence of Z δ <strong>and</strong> M δ we have alsop(x)‖x‖ ≤ b.By Proposition 4.4, we may assume that p is bounded by some constant K in B δ (e). Let‖x‖ > M δ .Choose a word w(x) = z 0 z 1 ...z n with ‖z i ‖ = δ(1 + ε i ) with |ε i | < ε, withp(x i ) < b‖x i ‖ = bδ(1 + ε i )<strong>and</strong>i.e.for some s with ‖s‖ < δ <strong>and</strong>d(x, w(x)) < δ,x = w(x)s1 − ε ≤ n(x)δ‖x‖ ≤ 1 + ε.
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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
- 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 and 58: Normed groups 53By (C-adm), we may
- Page 59 and 60: Normed groups 55equipped with an in
- 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: Normed groups 67The result confirms
- 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
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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,