130 N. H. Bingham <strong>and</strong> A. J. OstaszewskiProof. Referring to the Klee property, via the cyclic property we haved G (gag −1 , hbh −1 ) = ‖gag −1 hb −1 h −1 ‖ = ‖h −1 gag −1 hb −1 ‖≤ ‖h −1 g‖ + ‖ag −1 hb −1 ‖≤ ‖h −1 g‖ + ‖ab −1 ‖ + ‖g −1 h‖,for all a, b, yielding the right-h<strong>and</strong> side inequality. Then substitute g −1 ag for a etc., g −1for g etc., to obtaind G (a, b) ≤ 2d G (g −1 , h −1 ) + d G (gag −1 , hbh −1 ).This yields the left-h<strong>and</strong> side inequality, as d G is bi-invariant <strong>and</strong> sod G (g −1 , h −1 ) = ˜d G (g, h) = d G (g, h).Proposition 12.5 (Permutation metric). For π ∈ H(X), let d π (x, y) := d X (π(x), π(y)).Then d π is a metric, <strong>and</strong>d X (x, y) − 2‖π‖ ≤ d π (x, y) ≤ d X (x, y) + 2‖π‖.In particular, if d X is right-invariant <strong>and</strong> π(x) is the left-translation λ z (x) = zx, thend X (x, y) − 2‖z‖ ≤ d X z (x, y) = d X (zx, zy) ≤ d X (x, y) + 2‖z‖.Proof. By the triangle inequality,d X (π(x), π(y)) ≤ d X (π(x), x) + d X (x, y) + d X (y, π(y)) ≤ 2‖π‖ + d X (x, y).Likewise,d X (x, y) ≤ d X (x, π(x)) + d X (π(x), π(y)) + d X (π(y), y)≤ 2‖π‖ + d X (π(x), π(y)).If π(x) := zx, then ‖π‖ = sup d(zx, x) = ‖z‖ <strong>and</strong> the result follows.Recall from Proposition 2.2 that for d a metric on a group X, we write ˜d(x, y) =d(x −1 , y −1 ) for the (inversion) conjugate metric. The conjugate metric ˜d is left-invariantiff the metric d is right-invariant. Under such circumstances both metrics induce thesame norm (since d(e, x) = d(x −1 , e), as we have seen above). In what follows note thatξ −1x = ξ x −1.Theorem 12.6 (Quasi-isometric <strong>duality</strong>). If the metric d X on X is right-invariant <strong>and</strong>t ∈ T ⊂ H(X) is a subgroup, then˜d X (x, y) − 2‖t‖ H(X) ≤ d T (ξ x (t), ξ y (t)) ≤ ˜d X (x, y) + 2‖t‖ H(X) ,<strong>and</strong> hence, for each ε ≥ 0, the magnification metric (mag-eps) satisfies˜d X (x, y) − 2ε ≤ d ε T (ξ x , ξ y ) ≤ ˜d X (x, y) + 2ε.Equivalently, in terms of conjugate metrics,d X (x, y) − 2ε ≤ ˜d ε T (ξ x , ξ y ) ≤ d X (x, y) + 2ε.
<strong>Normed</strong> <strong>groups</strong> 131Hence,‖x‖ − 2ε ≤ ‖ξ x ‖ ε ≤ ‖x‖ + 2ε,<strong>and</strong> so ‖x n ‖ → ∞ iff d T (ξ x(n) (t), ξ e (t)) → ∞.Proof. We follow a similar argument to that for the permutation metric. By rightinvariance,sod X (t(x −1 z), t(y −1 z)) ≤ d X (t(x −1 z), x −1 z) + d X (x −1 z, y −1 z) + d X (y −1 z, t(y −1 z))≤ 2‖t‖ + d X (x −1 , y −1 ),d T (ξ x (t), ξ y (t)) = sup z d X (t(x −1 z), t(y −1 z)) ≤ 2‖t‖ + d X (x, e X ).Now, again by right-invariance,Butsoas required.d X (x −1 , y −1 ) ≤ d(x −1 , t(x −1 )) + d(t(x −1 ), t(y −1 )) + d(t(y −1 ), y −1 ).d(t(x −1 ), t(y −1 )) ≤ sup z d X (t(x −1 z), t(y −1 z)),d X (x −1 , y −1 ) ≤ 2‖t‖ + sup z d X (t(x −1 z), t(y −1 z)) = 2‖t‖ + d T (ξ x (t), ξ y (t)),We thus obtain the following result.Theorem 12.7 (Topological Quasi-Duality Theorem). For X a normed group, the seconddual Ξ is a normed group isometric to X which, for any ε ≥ 0, is ε-quasi-isometric to Xin relation to ˜d ε T (ξ x, ξ y ) <strong>and</strong> the ‖ · ‖ ε norm. Here T = H u (X).Proof. We metrize Ξ by setting d Ξ (ξ x , ξ y ) = d X (x, y). This makes Ξ an isometric copyof X <strong>and</strong> an ε-quasi-isometric copy in relation to the conjugate metric ˜d ε T (ξ x, ξ y ) whichis given, for any ε ≥ 0, byIn particular for ε = 0 we haved T (ξ −1x˜d ε T (ξ x , ξ y ) := sup ‖t‖≤ε d T (ξ −1x(t), ξy−1 (t)).(e), ξy−1 (e)) = sup z d X (xz, yz) = d(x, y).Assuming d X is right-invariant, d Ξ is right-invariant, sinced Ξ (ξ x ξ z , ξ y ξ z ) = d Ξ (ξ xz , ξ yz ) = d X (xz, yz) = d X (x, y).
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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
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Normed groups 715. Generic Dichotom
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Normed groups 73Returning to the cr
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Normed groups 75Examples. Here are
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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 and 126: Normed groups 121Proof. In the meas
- Page 127 and 128: Normed groups 123Hence, as t i n
- Page 129 and 130: Normed groups 125The corresponding
- Page 131 and 132: Normed groups 127(t, x) ✛✻Φ T
- Page 133: Normed groups 129Fix s. Since s is
- 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,