8 N. H. Bingham <strong>and</strong> A. J. OstaszewskiwhereB i ε(x) := {y ∈ X : d X i (x, y) < ε}.2. The strong ∆-refinement topology on X is defined by reference to the local base at xobtained by full intersections of ε-balls about x :⋂(Str)Clearly⋂d∈∆ Bd ε (x).d∈∆ Bd ε (x) ⊂ ⋂ i∈F Bi ε(x), for F finite,hence the name. We will usually be concerned with a family ∆ of conjugate metrics. Wenote the following, which is immediate from the definition. (For (ii) see the special casein [dGMc, Lemma 2.1], [Ru, Ch. I 1.38(c)], or [Eng, Th. 4.2.2 p. 259], which uses a sumin place of a supremum, <strong>and</strong> identify X with the diagonal of ∏ d∈∆(X, d); see also [GJ,Ch. 15].)Proposition 2.1. (i) The strong ∆-refinement topology is generated by the supremummetricd X ∆(x, y) = sup{d X i (x, y) : i ∈ I}.(ii) For ∆ a countable family of metrics indexed by I = N, the weak ∆-refinement topologyis generated by the weighted-supremum metricd X ∆(x, y) = sup i∈I 2 1−i d X i (x, y)1 + d X i (x, y).This corresponds to the metric of first-difference in a product of discrete metric spaces,e.g. in the additive group Z Z . (That is, d X ∆ ({x i}, {y i }) = 2 −n(x,y) , where the two sequencesfirst differ at index i = n(x, y).)Examples B. 1. For X a group we may take ∆ = {d X z<strong>and</strong> if d X is right-invariantd X ∆(x, y) = sup{d X (zx, zy) : z ∈ X},‖x‖ ∆ = sup z ‖zxz −1 ‖.: z ∈ X} to obtain2. For X a <strong>topological</strong> group we may take ∆ = {d X h: h ∈ Auth(X)}, to obtaind X ∆(x, y) = sup{d X (h(x), h(y)) : h ∈ Auth(X)}.3. In the case A = Auth(X) we may take ∆ = {d A x : x ∈ X}, the evaluation pseudometrics,to obtaind A ∆(f, g) = sup x d A x (f, g) = sup x d X (f(x), g(x)),‖f‖ ∆ = sup x d A x (f, id X ) = sup x d X (f(x), x).In Proposition 2.12 we will show that the strong ∆-refinement topology restricted to thesubgroup H(X) := {f ∈ A : ‖f‖ ∆ < ∞} is the topology of uniform convergence. Theweak ∆-refinement topology here is just the topology of pointwise convergence.<strong>and</strong>
<strong>Normed</strong> <strong>groups</strong> 9The following result is simple; we make use of it in the Definition which follows Lemma3.23.Proposition 2.2 (Symmetrization refinement). If ‖x‖ 0 is a group pre-norm, then thesymmetrization refinementis a group-norm.‖x‖ := max{‖x‖ 0 , ‖x −1 ‖ 0 }Proof. Positivity is clear, likewise symmetry. Noting that, for any A, B,a + b ≤ max{a, A} + max{b, B},<strong>and</strong> supposing without loss of generality thatmax{‖x‖ 0 + ‖y‖ 0 , ‖y −1 ‖ 0 + ‖x −1 ‖ 0 } = ‖x‖ 0 + ‖y‖ 0 ,we have‖xy‖ = max{‖xy‖ 0 , ‖y −1 x −1 ‖ 0 } ≤ max{‖x‖ 0 + ‖y‖ 0 , ‖y −1 ‖ 0 + ‖x −1 ‖ 0 }= ‖x‖ + ‖y‖ 0 ≤ max{‖x‖ 0 , ‖x −1 ‖ 0 } + max{‖y‖ 0 , ‖y −1 ‖ 0 }= ‖x‖ + ‖y‖.Remark. One can use summation <strong>and</strong> take ‖x‖ := ‖x‖ 0 + ‖x −1 ‖ 0 , as‖xy‖ = ‖xy‖ 0 + ‖y −1 x −1 ‖ 0 ≤ ‖x‖ 0 + ‖y‖ 0 + ‖y −1 ‖ 0 + ‖x −1 ‖ 0 = ‖x‖ + ‖y‖.However, here <strong>and</strong> below, we prefer the more general use of a supremum or maximum,because it corresponds directly to the intersection formula (Str) which defines the refinementtopology. We shall shortly see a further cogent reason (in terms of the refinementnorm).Proposition 2.3. If ‖ · ‖ is a group-norm, then d(x, y) := ‖xy −1 ‖ is a right-invariantmetric; equivalently, ˜d(x, y) := d(x −1 , y −1 ) = ‖x −1 y‖ is the conjugate left-invariant metricon the group.Conversely, if d is a right-invariant metric, then ‖x‖ := d(e, x) = ˜d(e, x) is a group-norm.Thus the metric d is bi-invariant iff ‖xy −1 ‖ = ‖x −1 y‖ = ‖y −1 x‖, i.e. iff the group-normis abelian.Furthermore, for (X, ‖ · ‖) a normed group, the inversion mapping x → x −1 from (X, d)to (X, ˜d) is an isometry <strong>and</strong> hence a homeomorphism.Proof. Given a group-norm put d(x, y) = ‖xy −1 ‖. Then ‖xy −1 ‖ = 0 iff xy −1 = e, i.e.iff x = y. Symmetry follows from inversion as d(x, y) = ‖(xy −1 ) −1 ‖ = ‖yx −1 ‖ = d(y, x).Finally, d obeys the triangle inequality, since‖xy −1 ‖ = ‖xz −1 zy −1 ‖ ≤ ‖xz −1 ‖ + ‖zy −1 ‖.
- Page 1 and 2: N. H. BINGHAM and A. J. OSTASZEWSKI
- Page 3 and 4: Normed groups 3ContentsContents . .
- Page 5 and 6: 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: Normed groups 74 (Topological permu
- 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 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.
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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].)
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Normed groups 79Remarks. 1. See [Fo
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Normed groups 81Theorem 6.1 (Catego
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Normed groups 83is continuous at th
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Normed groups 85compact. Evidently,
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Normed groups 87j ∈ ω} which enu
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Normed groups 89The result below ge
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Normed groups 91left-shift, not in
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Normed groups 93As a corollary of t
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Normed groups 953. For X a normed g
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Normed groups 97Proof. Note that‖
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Normed groups 99Taking h(x) := ‖
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Normed groups 1019. The Semigroup T
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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,