150 N. H. Bingham <strong>and</strong> A. J. Ostaszewski[Pet4] B. J. Pettis, Closed graph <strong>and</strong> open mapping theorems in certain <strong>topological</strong>lycomplete spaces, Bull. London Math. Soc. 6 (1974), 37–41.[Pic1] S. Piccard, Sur les ensembles de distances des ensembles de points d’unespace euclidien, Mém. Univ. Neuchâtel 13 (1939).[Pic2] S. Piccard, Sur des ensembles parfaites, Mém. Univ. Neuchâtel 16 (1942).[Pol]R. Pol, Remark on the restricted Baire property in compact spaces. Bull.Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 8, 599–603.[PWW] W. Poreda, <strong>and</strong> E. Wagner-Bojakowska <strong>and</strong> W. Wilczyński, A categoryanalogue of the density topology, Fund. Math. 125 (1985), no. 2, 167–173.[Rach] S. T. Rachev, Probability metrics <strong>and</strong> the stability of stochastic models,Wiley, 1991.[RR-01] K. P. S. Bhaskara Rao <strong>and</strong> M. Bhaskara Rao, A category analogue ofthe Hewitt-Savage zero-one law, Proc. Amer. Math. Soc. 44.2 (1974),497–499.[RR-TG] K. P. S. Bhaskara Rao <strong>and</strong> M. Bhaskara Rao, On the difference of twosecond category Baire sets in a <strong>topological</strong> group. Proc. Amer. Math.Soc. 47.1 (1975), 257–258.[Res] S. I. Resnick, Heavy-tail phenomena. Probabilistic <strong>and</strong> statistical modeling.Springer, New York, 2007.[Rog1] C. A. Rogers, Hausdorff measures, Cambridge University Press, 1970.[Rog2] C. A. Rogers, J. Jayne, C. Dellacherie, F. Topsøe, J. Hoffmann-Jørgensen,D. A. Martin, A. S. Kechris, A. H. Stone, Analytic sets, Academic Press,1980.[RW] C.A. Rogers, R. C. Willmott, On the projection of Souslin sets, Mathematika13 (1966), 147-150.[Ru]W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, 1991 (1st ed.1973).[Schw] L. Schwartz, Sur le théorème du graphe fermé, C. R. Acad. Sci. ParisSér. A-B 263 (1966), 602-605.[SeKu] I. E. Segal <strong>and</strong> R. A. Kunze, Integrals <strong>and</strong> operators, Mc-Graw-Hill,1968.[Se1] G. R. Sell, Nonautonomous differential equations <strong>and</strong> <strong>topological</strong> dynamics.I. The basic theory. Trans. Amer. Math. Soc. 127 (1967), 241-262.[Se2]G. R. Sell, Nonautonomous differential equations <strong>and</strong> <strong>topological</strong> dynamics.II. Limiting equations. Trans. Amer. Math. Soc. 127 (1967), 263–283.[She] S. Shelah, Can you take Solovay’s inaccessible away?, Israel J. Math. 48(1984), 1-47.[Si]M. Sion, Topological <strong>and</strong> measure-theoretic properties of analytic sets,Proc. Amer. Math. Soc. 11 (1960), 769–776.[SolSri] S. Solecki <strong>and</strong> S.M. Srivastava, Automatic continuity of group operations,Top. <strong>and</strong> Apps, 77 (1997), 65-75.
<strong>Normed</strong> <strong>groups</strong> 151[So]R. M. Solovay, A model of set theory in which every set of reals isLebesgue measurable, Ann. of Math. 92 (1970), 1-56.[St]H. Steinhaus, Sur les distances des points de mesure positive, Fund.Math. 1(1920), 93-104.[T] F. D. Tall, The density topology, Pacific J. Math. 62.1 (1976), 275–284.[TV]T. Tao, V.N. Vu, Additive combinatorics, Cambridge University Press,2006.[THJ] F. Topsøe, J. Hoffmann-Jørgensen, Analytic spaces <strong>and</strong> their applications,Part 3 of [Rog2].[Ul] S. M. Ulam, A collection of mathematical problems, Wiley, 1960.[Ung] G. S. Ungar, On all kinds of homogeneous spaces, Trans. Amer. Math.Soc. 212 (1975), 393–400.[vD]E. van Douwen, A technique for constructing honest locally compactsubmetrizable examples. Topology Appl. 47 (1992), no. 3, 179–201.[vM1] J. van Mill, A note on the Effros Theorem, Amer. Math. Monthly 111.9(2004), 801-806.[vM2] J. van Mill, The topology of homeomorphism <strong>groups</strong>, in preparation,partly available at:http://www.cs.vu.nl/˜dijkstra/teaching/Caput/<strong>groups</strong>.pdf[Var] N. Th.. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis <strong>and</strong> geometryon <strong>groups</strong>, Cambridge Tracts in Mathematics 100, Cambridge UniversityPress, 1992.[vdW] B. L. van der Waerden, Modern Algebra. Vol. I., Ungar Publishing Co.,New York, 1949.[We]A. Weil, L’intégration dans les groupes topologiques et ses applications,Actual. Sci. Ind., no. 869. Hermann et Cie., Paris, 1940 [republished,Princeton, N. J., 1941].[WKh] W. Wilczyński <strong>and</strong> A. B. Kharazishvili, Translations of measurable sets<strong>and</strong> sets having the Baire property, Soobshch. Akad. Nauk Gruzii, 145.1(1992), 43-46.[Wr] F.B. Wright, Semi<strong>groups</strong> in compact <strong>groups</strong>, Proc. Amer. Math. Soc. 7(1956), 309–311.[Ya]A. L.Yakimiv, Probabilistic applications of Tauberian theorems, VSP,Leiden, 2005.[Zel]W. Żelazko, A theorem on B 0 division algebras, Bull. Acad. Polon. Sci.8(1960), 373-375.
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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
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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‖
- 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: Normed groups 149fields. I. Basic p