Normed versus topological groups: Dichotomy and duality

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142 N. H. Bingham and A. J. Ostaszewski(1958), 648–652.[Berg] C. Berge, Topological spaces, including a treatment of multi-valued functions,vector spaces and convexity, Engl. Translation, Oliver and Boyd,1963 (reprint, Dover 1997).[BHW] V. Bergelson, N. Hindman, B. Weiss, All-sums sets in (0,1] – categoryand measure, Mathematika 44 (1997), no. 1, 61–87.[Berz] E. Berz, Sublinear functions on R. Aequationes Math. 12, no. 2/3 (1975),200–206.[BePe] Cz. Bessaga and A. Pe̷lczyński, Selected topics in infinite-dimensionaltopology, PWN, 1975.[BG]N. H. Bingham, C.M. Goldie, Extensions of regular variation, I: Uniformityand quantifiers, Proc. London Math. Soc. (3) 44 (1982), 473-496.[BGT] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, 2nd edition,Encycl. Math. Appl. 27, Cambridge University Press, Cambridge,1989 (1st edition 1987).[BOst-GenSub] N. H. Bingham and A. J. Ostaszewski, Generic subadditive functions,Proc. Amer. Math. Soc. 136 (2008), 4257-4266.[BOst-FRV] N. H. Bingham and A. J. Ostaszewski, Infinite Combinatorics and thefoundations of regular variation, J. Math. Anal. Appl. 360 (2009), 518-529.[BOst-Funct] N. H. Bingham and A. J. Ostaszewski, Infinite combinatorics in functionspaces: category methods, Publ. Inst. Math. Béograd, 86 (100) (2009),55-73.[BOst-LBI] N. H. Bingham and A. J. Ostaszewski, Beyond Lebesgue and Baire:generic regular variation, Colloquium Mathematicum, 116.1 (2009), 119-138.[BOst-Aeq] N. H. Bingham and A. J. Ostaszewski, Automatic continuity: subadditivity,convexity, uniformity, Aequationes Math. 78 (2009), 257-270.[BOst-Thin] N. H. Bingham and A. J. Ostaszewski, Automatic continuity via analyticthinning, Proc. Amer. Math. Soc. 138 (2010), 907-919.[BOst-RVWL] N. H. Bingham and A. J. Ostaszewski, Regular variation without limits,Journal of Math. Anal. Appl. 370 (2010), 322-338.[BOst-LBII] N. H. Bingham and A. J. Ostaszewski, Beyond Lebesgue and Baire II:Bitopology and measure-category duality, Colloquium Math., in press.[BOst-TRI] N. H. Bingham and A. J. Ostaszewski, Topological regular variation: I.Slow variation, Topology and App. 157(2010), 1999-2013.[BOst-TRII] N. H. Bingham and A. J. Ostaszewski, Topological regular variation: II.The fundamental theorems, Topology and App. 157(2010), 2014-2023.[BOst-TRIII] N. H. Bingham and A. J. Ostaszewski, Topological regular variation: III.Regular variation, Topology and App. 157(2010), 2024-2037.[BOst-KCC] N. H. Bingham and A. J. Ostaszewski, Kingman, category and combinatorics,Probability and mathematical genetics (J.F.C. KingmanFestschrift, ed. N.H. Bingham and C.M. Goldie), LMS Lecture Notes
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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,
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Normed groups 87j ∈ ω} which enu
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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: Normed groups 141Bibliography[AL]J.
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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