6 N. H. Bingham <strong>and</strong> A. J. OstaszewskiDefinition <strong>and</strong> Notation. For X a metric space with metric d X <strong>and</strong> π : X → X abijection the π-permutation metric is defined byWhen X is a group we will also say that d X π<strong>and</strong> for d any metric on Xd X π (x, y) := d X (π(x), π(y)).‖x‖ π := d X (π(x), π(e)),B d r (x) := {y : d(x, y) < r},suppressing the superscript for d = d X ; however, for d = d X πB π r (x) := {y : d X π (x, y) < r}.is the π-conjugate of d X . We writewe adopt the briefer notationFollowing [BePe] Auth(X) denotes the algebraic group of self-homeomorphisms (or autohomeomorphisms)of X under composition, i.e. without a <strong>topological</strong> structure. We denoteby id X the identity map id X (x) = x on X.Examples A. Let X be a group with metric d X . The following permutation metricsarise naturally in this study. (We use the notation ‖x‖ := d X (x, e X ), for an arbitrarymetric.)1. With π(x) = x −1 we refer to the π-permutation metric as the involution-conjugate, orjust the conjugate, metric <strong>and</strong> write˜d X (x, y) = d X π (x, y) = d X (x −1 , y −1 ), so that ‖x‖ π = ‖x −1 ‖.2. With π(x) = γ g (x) := gxg −1 , the inner automorphism, we have (dropping the additionalsubscript, when context permits):d X γ (x, y) = d X (gxg −1 , gyg −1 ), so that ‖x‖ γ = ‖gxg −1 ‖.3. With π(x) = λ g (x) := gx, the left-shift by g, we refer to the π-permutation metric asthe g-conjugate metric, <strong>and</strong> we writed X g (x, y) = d X (gx, gy).If d X is right-invariant, cancellation on the right givesd X (gxg −1 , gyg −1 ) = d X (gx, gy), i.e. d X γ (x, y) = d X g (x, y) <strong>and</strong> ‖x‖ g = ‖gxg −1 ‖.For d X right-invariant, π(x) = ρ g (x) := xg, the right-shift by g, gives nothing new:But, for d X left-invariant, we haved X π (x, y) = d X (xg, yg) = d X (x, y).‖x‖ π = ‖g −1 xg‖.
<strong>Normed</strong> <strong>groups</strong> 74 (Topological permutation). For π ∈ Auth(X), i.e. a homeomorphism <strong>and</strong> x fixed,note that for any ε > 0 there is δ = δ(ε) > 0 such thatprovided d(x, y) < δ, i.e.d π (x, y) = d(π(x), π(y)) < ε,B δ (x) ⊂ B π ε (x).Take ξ = π(x) <strong>and</strong> write η = π(y); there is µ > 0 such thatd(x, y) = d π −1(ξ, η) = d(π −1 (ξ), π −1 (η)) < ε,provided d π (x, y) = d(π(x), π(y)) = d(ξ, η) < µ, i.e.B π µ(x) ⊂ B ε (x).Thus the topology generated by d π is the same as that generated by d. This observationapplies to all the previous examples provided the permutations are homeomorphisms (e.g.if X is a <strong>topological</strong> group under d X ). Note that for d X right-invariant‖x‖ π = ‖π(x)π(e) −1 ‖.5. For g ∈ Auth(X), h ∈ X, the bijection π(x) = g(ρ h (x)) = g(xh) is a homeomorphismprovided right-shifts are continuous. We refer to this as the shifted g-h-permutation metricwhich has the associated g-h-shifted normd X g-h(x, y) = d X (g(xh), g(yh)),‖x‖ g-h = d X (g(xh), g(h)).6 (Equivalent Bounded norm). Set d b (x, y) = min{d X (x, y), 1}. Then d b is an equivalentmetric (cf. [Eng, Th. 4.1.3, p. 250]). We refer toas the equivalent bounded norm.‖x‖ b := d b (x, e) = min{d X (x, e), 1} = min{‖x‖, 1}7. For A = Auth(X) the evaluation pseudo-metric at x on A is given by<strong>and</strong> sois a pseudo-norm.d A x (f, g) = d X (f(x), g(x)),‖f‖ x = d A x (f, id) = d X (f(x), x)Definition (Refinements). 1 (cf. [GJ, Ch. 15.3] which works with pseudometrics). Let∆ = {d X i : i ∈ I} be a family of metrics on a group X. The weak (Tychonov) ∆-refinementtopology on X is defined by reference to the local base at x obtained by finite intersectionsof ε-balls about x :⋂i∈F Bi ε(x), for F finite, i.e. Bε i1(x) ∩ ... ∩ Bε in(x), if F = {i 1 , ..., i n },
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- 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: Normed groups 5abelian group has se
- Page 13 and 14: Normed groups 9The following result
- 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
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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].)
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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,