Normed versus topological groups: Dichotomy and duality

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Indexabelian, 4Abelian Normability of H(X), 39admissibilityCauchy (C-adm), 51metric, 18norm, 18, 37topological , 18admissibilityweak (W-adm), 51almost complete, 72ambidextrous refinement, 32ambidextrous uniformity, 10, 54amenable, 5analytic, 112Analytic Automaticity Th., 125Analytic Dichotomy Lemma – Spanning, 119asymptotically multiplicative, 21Auth(X), 6Averaging Lemma, 104Baire, 49Baire Continuity Th., 115Baire function, 114Baire Homomorphism Th., 117Baire-continuous, 114Baire-measurable, 114Banach-Kuratowski Th., 88Banach-Mehdi Th., 116Banach-Neeb Theorem, 115Bartle’s Inverse Function Th., 111Bernstein-Doetsch Th., 107bi-Cauchy, 49Bi-Cauchy completion, 54bi-invariant (not) – SL(2, R), 38Bi-Lipschitz property, 43bi-uniformly continuous, 33, 126Borel/analytic inversion, 56bounded, 20bounded elements, 15[138]C-sets, 114Cauchy dichotomy, 49centre, 44Z γ (G), norm centre, 44CET-Category Embedding Th., 81Choquet’s Capacitability Theorem, 120class C ′ , 110closed graph, 46Combinatorial Uniform Boundedness Th.,95Common Basis Th., 101commutator, 23Compact Contraction Lemma, 120Compact Spanning Approximation, 121compact-open topology, 17completely metrizable, 71conjugacy refinement norm, 17Conjunction Th., 102converges to the identity, 133, 135convex, 104Crimping Th., 35d X R , 5Darboux norm, 123Darboux’s Th., 49Darboux-normed , 55Dense Oscillation Th., 62Density Topology Th., 91density-preserving, 79derivative, 110Dichotomy Th., convex functions, 108Displacements Lemma – Baire, 75Displacements Lemma – measure, 76diverges uniformly, 133divisible – 2-divisible, 103divisible – infinitely divisible, 14ε-shifting point, 61Effros’ Open Mapping Principle, 35
Normed groups 139embeddable, 14enables Baire continuity, 115ε-swelling, 11Equivalence Th., 29equivalent bounded norm, 7evaluation map, 31Example C, 18Examples A., 6Examples B, 8First Verification Th., 82g-conjugate norm, 10G-ideal, 124γ g (x), 6Generalized Darboux Th., 123Generalized Mehdi Th., 111Generalized Piccard-Pettis Th., 89Generic Dichotomy Principle, 74Global Bounds Th., 97Group of left shifts, 33group-norm, 4Heine-Borel property , 57Heine-Borel Th., 50N-homogeneous, 41homogeneous – 2-homogeneous, 5homogeneous – n-homogeneous, 41homomorphism, 27infinitely divisible, 40inner-regularity, 72Interior Point Th., 99Invariance of Norm Th., 22Jones-Kominek Th. , 122Kakutani-Birkhoff Th., 23KBD – First Generalized Measurable Th.,93KBD – Kestelman-Borwein-Ditor Th., 2KBD – normed groups, 2KBD – Second Generalized Measurable, 119KBD – topological groups, 83KBD – topologically complete norm, 72Klee group, 26Klee property, 22, 30Kodaira’s Th., 93Kuratowski Dichotomy, 88left-invariant metric, 9Left-right Approximation, 63left-right commutator inequality, 25λ g (x), 6Lindelöf, 112Lipschitz properties of H u , 128Lipschitz property, 40Lipschitz-1 norms, 40Lipschitz-normed, 40locally bi-Lipschitz, 105locally convex, 108locally Lipschitz, 105lower hull, 67Luzin hierarchy, 114Magnification metric, 21McShane’s Interior Points Th., 90meagre, 49modular, 105Montgomery’s Th., 61multiplicative, 21N-homogeneous – homomorphism, 123n-Lipschitz, 105nearly abelian norm, 46Nikodym’s Th., 114No Trumps Th., 95norm topology, 26norm-central, 44oscillation function, 57Pathology Th., 65Permutation metric, 130Piccard Th., 100Piccard-Pettis Th., 84pointwise divergent sequence, 133Product Set Th., 103Q-good, 125quasi-continuous, 66Quasi-isometric duality, 130
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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
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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,
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: Normed groups 137However, whilst th
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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