106 N. H. Bingham <strong>and</strong> A. J. Ostaszewskiis injective. If, moreover, the group is n-divisible, then the power function f n (x) is anisomorphism.We note that in the additive group of reals x 2 fails to be locally bi-Lipschitz atthe origin (since its derivative there is zero): see [Bart]. However, the following are bi-Lipschitz. 1. In R d with additive notation, we have ‖x 2 ‖ := ‖2x‖ = 2‖x‖, so the norm is2-Lipschitz. 2. In R ∗ + we have ‖x 2 ‖ := | log x 2 | = 2| log x| = 2‖x‖ <strong>and</strong> again the norm is2-Lipschitz. 3. In a Klee group the mapping f(x) := x n is uniformly (locally) Lipschitz,since∣ ∣ ∣x n y −n∣ ∣ ∣ G≤ n‖xy −1 ‖ G ,proved inductively from the Klee property (Th. 2.18) via the observation that∣∣ ∣∣∣x n+1 y −(n+1) ∣∣ ∣∣G= ∣ ∣ xx n y −n y −1∣ ∣ ∣ G≤ ∣ ∣ x n y −n∣ ∣ ∣ G+ ∣ ∣ ∣ xy−1 ∣ G.Lemma 10.4 (Reflecting Lemma). Suppose the group-norm is everywhere locally 2-Lipschitz. Then, for T non-meagre, T is reflecting i.e. there are w ∈ G (a right-reflectingtranslator) <strong>and</strong> {v n } ⊆ T such that, for infinitely many n ∈ ω, we haveThere is likewise a left-reflecting translator.v 2 n = u n w.Proof. Let T 2 := {g : g = t 2 for some t ∈ T }. By assumption, T 2 is non-meagre. Withu n = uz n , put S = T 2 <strong>and</strong> notice that u n w ∈ S iff uz n w ∈ S iff z n w ∈ u −1 S. Now u −1 Sis non-meagre <strong>and</strong> ψ n (x) := z n x as usual converges to the identity, so the existence ofw ∈ u −1 S is assured such that z n w = u −1 v 2 n.Remarks. 1. Note that the assertion here isso thatord(v n , w) = d(v −1nu −1n v n = wvn −1 ,, u −1n ) = ˜d(v n , u n ) ≈ ˜d(v n , u),d(v n , w) ≈ ˜d(v n , u),suggesting the terminology of reflection.2. Boundedness theorems for reflecting <strong>and</strong> averaging sets follow as in [BOst-Aeq] sincethe following are true in any group, as we see below.Theorem 10.5. In a normed <strong>topological</strong> group, for f a 1 2-convex function, if f is locallybounded above at x 0 , then it is locally bounded below at x 0 (<strong>and</strong> hence locally bounded atx 0 ).
<strong>Normed</strong> <strong>groups</strong> 107Proof. Say f is bounded above in B := B δ (x 0 ) by M. Consider u ∈˜d(x 0 , u) = ‖u −1 x 0 ‖ < δ. Put t = u −1 x 2 0; then tx −10 = u −1 x 0 , <strong>and</strong> soThen t ∈ B, <strong>and</strong> since x 2 0 = ut we haveord(t, x 0 ) = ‖tx −10 ‖ = ‖u−1 x 0 ‖ = ˜d(u, x 0 ) < δ.2f(x 0 ) ≤ f(u) + f(t) ≤ f(u) + M,f(u) ≥ 2f(x 0 ) − M.Thus 2f(x 0 ) − M is a lower bound for f on the open set ˜B δ (x 0 ).˜B δ (x 0 ). ThusAs a corollary a suitably rephrased Bernstein-Doetsch Theorem ([Kucz], [BOst-Aeq])is thus true.Theorem 10.6 (Bernstein-Doetsch Theorem). In a normed group, for fa 1 2-convex function,if f is locally bounded above at x 0 , then f is continuous at at x 0 .Proof. We repeat the ‘Second proof’ of [Kucz, p. 145]. Choose y n → x 0 with f(y n ) →m f (x 0 ) <strong>and</strong> z n → x 0 with f(z n ) → M f (x 0 ). Let u n :=ynx 2 −1n . Thus yn 2 = u n x n <strong>and</strong> so2f(y n ) ≤ f(u n ) + f(z n ),i.e. f(u n ) ≥ 2f(y n ) − f(z n ). Hence in the limit we obtainM f (x 0 ) ≥ lim inf f(u n ) ≥ 2M f (x 0 ) − m f (x 0 ).One thus has that M f (x 0 ) ≤ m f (x 0 ). But m f (x 0 ) ≤ f(x 0 ) ≤ M f (x 0 ), <strong>and</strong> both hullvalues are finite (by the result above). Thus m f (x 0 ) = f(x 0 ) = M f (x 0 ), from whichcontinuity follows.We now consider the transferability of upper <strong>and</strong> lower local boundedness. Our proofswork directly with definitions (so are not modelled after those in Kuczma [Kucz]). Wedo not however consider domains other than the whole metric group. For clarity of proofstructure we give separate proofs for the two cases, first when Gis abelian <strong>and</strong> later forgeneral G.Theorem 10.7 (Local upper boundedness). In a normed <strong>topological</strong> group G, for fa12 -convex function defined on G, if f is locally bounded above at some point x 0, then f islocally bounded above at all points.Proof. Case (i) The Abelian case. Say f is bounded above in B := B δ (x 0 ) by M. Givena fixed point t, put z = z t := x −10 t2 , so that t 2 = x 0 z. Consider any u ∈ B δ/2 (t).Write u = st with ‖s‖ < δ/2. Now put y = s 2 ; then ‖y‖ = ‖s 2 ‖ ≤ 2‖s‖ < δ. Henceyx 0 ∈ B δ (x 0 ). Nowu 2 = (st) 2 = s 2 t 2 = yx 0 z,
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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
- 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 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
- Page 109: Normed groups 105By the Category Em
- 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 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,