Normed versus topological groups: Dichotomy and duality

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N. H. BinghamDepartment of MathematicsImperial CollegeSouth KensingtonLondon SW7 2AZ, UKE-mail: nick.bingham@imperial.ac.ukA. J. OstaszewskiDepartment of MathematicsLondon School of EconomicsHoughton StreetLondon WC2A 2AE, UKE-mail: a.j.ostaszewski@lse.ac.uk
Normed groups 3ContentsContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Metric versus normed groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. Normed versus topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1. Left versus right-shifts: Equivalence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2. Lipschitz-normed groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3. Cauchy Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494. Subadditivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675. Generic Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716. Steinhaus theory and Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807. The Kestelman-Borwein-Ditor Theorem: a bitopological approach . . . . . . . . . . . . . . . 918. The Subgroup Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999. The Semigroup Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10010. Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10311. Automatic continuity: the Jones-Kominek Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11212. Duality in normed groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12613. Divergence in the bounded subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Page 1: N. H. BINGHAM and A. J. OSTASZEWSKI
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 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.
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 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 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,
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Normed versus topological groups: Dichotomy and duality

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