Normed versus topological groups: Dichotomy and duality

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140 N. H. Bingham and A. J. Ostaszewskiquasi-isometry, 10, 96Ramsey’s Th., 50refinement norm, 10refinement topology, 7Reflecting Lemma, 106right-invariant metric, 9Right-invariant sup-norm, 18ρ g (x), 6right-shift compact, 84Second Verification Th., 92Self-similarity Th., 102semicontinuous, 70semicontinuous – lower, 46semicontinuous – upper, 46Semigroup Th., 101, 103semitopological, 60semitopological group, 26sequence space, 54, 124sequential, 125sequential – completely sequential, 125Shift-Compactness Th., 85shifted-cover, 86slowly varying, 12smooth, 110Souslin criterion - Baire functions, 116Souslin hierarchy, 114Souslin-H, 113Souslin-graph Th., 118span, 113Squared Pettis Th., 78Steinhaus Th., 100Steinhaus Th. – Weil Topology, 87subadditive, 67subcontinuous, 50Subgroup Dichotomy Th. – normed groups,88Subgroup Dichotomy Th. – topological groups,88Subgroup Th., 100subuniversal set, 94supremum norm, 15Th. of Jones and Kominek, 112thick, 84topological centre, 59Topological Quasi-Duality Th., 131topological under weak refinement, 16topologically complete, 71unconditional divergence, 136Ungar’s Th., 36Uniformity Th. for Conjugation, 129uniformly continuous, 52uniformly divergent sequence, 133vanishingly small word-net , 41weak category convergence, 80weak continuity, 29, 52
Normed groups 141Bibliography[AL]J. M. Aarts and D. J. Lutzer, Completeness properties designed for recognizingBaire spaces. Dissertationes Math. (Rozprawy Mat.) 116 (1974),48.[AD]J. Aczél and J. Dhombres, Functional equations in several variables, Encycl.Math. Appl. 31, Cambridge University Press, Cambridge, 1989.[AdC] O. Alas and A. di Concilio, Uniformly continuous homeomorphisms,Topology Appl. 84 (1998), no. 1-3, 33–42.[AS]N. Alon and J. H. Spencer, The probabilistic method, 3rd ed., Wiley,2008 (2nd. ed. 2000, 1st ed. 1992).[AnB] R. D. Anderson and R. H. Bing, A completely elementary proof thatHilbert space is homeomorphic to the countable infinite product of lines,Bull. Amer. Math. Soc. 74 (1968), 771-792.[Ar1] R. F. Arens, A topology for spaces of transformations, Ann. of Math. (2)47, (1946). 480–495.[Ar2] R. F. Arens, Topologies for homeomorphism groups, Amer. J. Math. 68,(1946). 593–610.[ArMa] A.V. Arkhangel’skii, V.I. Malykhin, Metrizability of topological groups.(Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1996, no. 3, 13–16,91; translation in Moscow Univ. Math. Bull. 51 (1996), no. 3, 9–11.[ArRez] A.V. Arkhangel’skii, , E. A. Reznichenko, Paratopological and semitopologicalgroups versus topological groups, Topology and its App. 151(2005), 107-119.[Ba1] R. Baire, Thèse: Sur les fonctions de variable réelle, Ann. di Math. (3),3 (1899), 65.[Ba2] R. Baire, Sur la representation des fonctions discontinues (2me partie),Acta Math. 32 (1909), 97-176.[BajKar] B. Bajšanski and J. Karamata, Regular varying functions and the principleof equicontinuity, Publ. Ramanujan Inst. 1 (1969) 235-246.[Ban-Eq] S. Banach, Sur l’équation fonctionelle f(x + y) = f(x) + f(y), Fund.Math. 1(1920), 123-124, reprinted in collected works vol. I, 47-48, PWN,Warszawa, 1967 (Commentary by H. Fast p. 314).[Ban-G] S. Banach, Über metrische Gruppen, Studia Math. III (1931), 101-113,reprinted in Collected Works vol. II, 401-411, PWN, Warsaw, 1979.[Ban-T] S. Banach, Théorie des opérations linéaires, reprinted in Collected Worksvol. II, 401-411, PWN, Warsaw, 1979 (1st. edition 1932).[Bart] R. G. Bartle, Implicit functions and solutions of equations in groups,Math. Z. 62 (1955), 335–346.[Be]A. Beck, A note on semi-groups in a locally compact group. Proc. Amer.Math. Soc. 11 (1960), 992–993.[BCS] A. Beck, H.H, Corson, A. B. Simon, The interior points of the productof two subsets of a locally compact group, Proc. Amer. Math. Soc. 9
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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
Page 19 and 20:
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,
Page 29 and 30:
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,
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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: Normed groups 139embeddable, 14enab
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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