12 N. H. Bingham <strong>and</strong> A. J. OstaszewskiFor further information on norms with the Heine-Borel property (for which compactsets are precisely those sets which are closed in the right norm topology <strong>and</strong> normbounded)see [?]).The significance of the following simple corollary is wide-ranging. It explicitly demonstratesthat small either-sided translations λ x , ρ y do not much alter the norm. Its maineffect is on the analysis of subadditive functions.Corollary 2.6. With ‖x‖ := d X (x, e), where d X is a right-invariant metric on X,|(‖x‖ − ‖y‖)| ≤ ‖xy‖ ≤ ‖x‖ + ‖y‖.Proof. By Proposition 2.2, the triangle inequality <strong>and</strong> symmetry holds for norms, so‖y‖ = ‖x −1 xy‖ ≤ ‖x −1 ‖ + ‖xy‖ = ‖x‖ + ‖xy‖.We now generalize (rv-limit), by letting T, X be sub<strong>groups</strong> of a normed group G withX invariant under T.Definition. We say that a function h : X → H is slowly varying on X over T if∂ X h(t) = e H , that is, for each t in Th(tx)h(x) −1 → e H , as ‖x‖ → ∞ for x ∈ X.We omit mention of X <strong>and</strong> T when context permits. In practice G will be an internaldirect product of two normal sub<strong>groups</strong> G = T X. (For a <strong>topological</strong> view on the internaldirect product, see [Na, Ch. 2.7] ; for an algebraic view see [vdW, Ch. 6, Sect. 47], [J] Ch. 9<strong>and</strong> 10, or [Ga] Section 9.1.) We may verify the property of h just defined by comparisonwith a slowly varying function.Theorem 2.7 (Comparison criterion). h : X → H is slowly varying iff for some slowlyvarying function g : X → H <strong>and</strong> some µ ∈ H,lim ‖x‖→∞ h(x)g(x) −1 = µ.Proof. If this holds for some slowly varying g <strong>and</strong> some µ,h(tx)h(x) −1 = h(tx)g(tx) −1 g(tx)g(x) −1 g(x)h(x) −1 → µe H µ −1 = e H ,so h is slowly varying; the converse is trivial.Theorem 2.8. For d X a right-invariant metric on a group X, the norm ‖x‖ := d X (x, e),as a function from X to the multiplicative positive reals R ∗ +, is slowly varying in themultiplicative sense, i.e., for any t ∈ X,Hence alsolim ‖x‖→∞‖tx‖‖x‖ = 1.lim ‖x‖→∞‖gxg −1 ‖‖x‖= 1.
<strong>Normed</strong> <strong>groups</strong> 13More generally, for T a one-parameter subgroup of X, any sub-additive Baire functionp : X → R ∗ + with‖p‖ T := lim x∈T, ‖x‖→∞p(x)‖x‖ > 0is multiplicatively slowly varying. (The limit exists by the First Limit Theorem for Bairesubadditive functions, see [BOst-GenSub].)Proof. By Corollary 2.6, for x ≠ e,1 − ‖t‖‖x‖ ≤ ‖tx‖‖x‖ ≤ 1 + ‖t‖‖x‖ ,which implies slow variation. We regard p as mapping to R ∗ +, the strictly positive reals(since p(x) = 0 iff x = e X ). Taking h = p <strong>and</strong> µ = ‖p‖ T > 0, the assertion follows fromthe Comparison Criterion (Th. 2.7) above (with g(x) = ‖x‖). Explicitly, for x ≠ e,p(xy)p(x) = p(xy)‖xy‖ · ‖xy‖‖x‖ · ‖x‖p(x) → ‖p‖ 1T · 1 · = 1.‖p‖ TCorollary 2.9. If π : X → Y is a group homomorphism <strong>and</strong> ‖ · ‖ Y is (1-γ)-quasiisometricto ‖ · ‖ X under the mapping π, then the subadditive function p(x) = ‖π(x)‖ Yis slowly varying. For general (µ-γ)-quasi-isometry the function p satisfieswhereµ −2 ≤ p ∗ (z) ≤ p ∗ (z) ≤ µ 2 ,p ∗ (z) = lim sup ‖x‖→∞ p(zx)p(x) −1 p ∗ (z) = lim inf ‖x‖→∞ p(zx)p(x) −1 .Proof. Subadditivity of p follows from π being a homomorphism, since p(xy) = ‖π(xy)‖ Y= ‖π(x)π(y)‖ Y ≤ ‖π(x)‖ Y + ‖π(y)‖ Y . Assuming that, for µ = 1 <strong>and</strong> γ > 0, the norm‖ · ‖ Y is (µ-γ)-quasi-isometric to ‖ · ‖ X , we have, for x ≠ e,So1 − γ ≤ p(x) ≤ 1 −γ .‖x‖ X ‖x‖ X ‖x‖ Xlim ‖x‖→∞p(x)‖x‖ = 1 ≠ 0,<strong>and</strong> the result follows from the Comparison Criterion (Th. 2.7) <strong>and</strong> Theorem 2.5.If, for general µ ≥ 1 <strong>and</strong> γ > 0, the norm ‖ · ‖ Y is (µ-γ)-quasi-isometric to ‖ · ‖ X , wehave, for x ≠ e,µ −1 − γ ≤ p(x) ≤ µ −γ .‖x‖ X ‖x‖ X ‖x‖ XSo for y fixedp(xy)p(x) = p(xy)‖xy‖ · ‖xy‖‖x‖ · ‖x‖p(x) ≤ (µ − γ‖xy‖ X)· ‖xy‖‖x‖ ·(µ −1 − γ‖x‖ X) −1,
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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 and 10: Normed groups 5abelian group has se
- Page 11 and 12: Normed groups 74 (Topological permu
- Page 13 and 14: Normed groups 9The following result
- Page 15: Normed groups 11Corollary 2.4. 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.
- Page 63 and 64: Normed groups 59Definition. A point
- Page 65 and 66: 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,