148 N. H. Bingham <strong>and</strong> A. J. Ostaszewskipotential theory, Lecture Notes in Mathematics 1189, Springer, 1986.[Lyn1] R. C. Lyndon, Equations in free <strong>groups</strong>. Trans. Amer. Math. Soc. 96(1960), 445–457.[Lyn2] R. C. Lyndon, Length functions in <strong>groups</strong>, Math. Sc<strong>and</strong>. 12 (1963), 209–234[LynSch] R. C. Lyndon <strong>and</strong> P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.[Mar] N.F.G. Martin, A topology for certain measure spaces, Trans. Amer.Math. Soc. 112 (1964), 1–18.[McCN] R. A. McCoy, I. Ntantu, Topological properties of spaces of continuousfunctions, Lecture Notes in Mathematics, 1315. Springer-Verlag, Berlin,1988.[McC] M. McCrudden, The embedding problem for probabilities on locally compact<strong>groups</strong>, Probability measures on <strong>groups</strong>: recent directions <strong>and</strong> trends(ed. S.G. Dani <strong>and</strong> P. Graczyk), 331–363, Tata Inst. Fund. Res., Mumbai,2006.[McSh] E. J. McShane, Images of sets satisfying the condition of Baire, Ann.Math. 51.2 (1950), 380-386.[MeSh] M.S. Meerschaert <strong>and</strong> H.-P. Scheffler, Limit distributions for sums ofindependent r<strong>and</strong>om vectors: Heavy tails in theory <strong>and</strong> Practice, Wiley,2001.[Meh] M.R. Mehdi, On convex functions, J. London Math. Soc. 39 (1964), 321-326.[Michael] E. Michael, Almost complete spaces, hypercomplete spaces <strong>and</strong> relatedmapping theorems, Topology Appl. 41.1 (1991), 113–130.[Michal1] A. D. Michal, Differentials of functions with arguments <strong>and</strong> values in<strong>topological</strong> abelian <strong>groups</strong>. Proc. Nat. Acad. Sci. U. S. A. 26 (1940),356–359.[Michal2] A. D. Michal, Functional analysis in <strong>topological</strong> group spaces, Math.Mag. 21 (1947), 80–90.[Mil]J. Milnor, A note on curvature <strong>and</strong> fundamental group. J. DifferentialGeometry 2 (1968), 1–7.[Mont0] D. Montgomery, Nonseparable metric spaces, Fund. Math.25 (1935), 527-534.[Mon1] D. Montgomery, Continuity in <strong>topological</strong> <strong>groups</strong>, Bull. Amer. Math.Soc. 42 (1936), 879-882.[Mon2] D. Montgomery, Locally homogeneous spaces, Ann. of Math. (2) 52(1950), 261–271.[Mue] B. J. Mueller, Three results for locally compact <strong>groups</strong> connected withthe Haar measure density theorem, Proc. Amer. Math. Soc. 16.6 (1965),1414-1416.[Na] L. Nachbin, The Haar integral, Van Nostr<strong>and</strong>, 1965.[NSW] A. Nagel, E. M. Stein, S. Wainger, Balls <strong>and</strong> metrics defined by vector
<strong>Normed</strong> <strong>groups</strong> 149fields. I. Basic properties, Acta Math. 155 (1985), 103–147.[Nam] I. Namioka, Separate <strong>and</strong> joint continuity, Pacific J. Math. 51 (1974),515–531.[Ne] K.-H. Neeb, On a theorem of S. Banach, J. Lie Theory 7 (1997), no. 2,293–300.[Nik] O. Nikodym, Sur une propriété de l’opération A, Fund. Math. 7 (1925),149-154.[NR]A. Nowik <strong>and</strong> P. Reardon, A dichotomy theorem for the Ellentuck topology,Real Anal. Exchange 29 (2003/04), no. 2, 531–542.[OC]W. Orlicz, <strong>and</strong> Z. Ciesielski, Some remarks on the convergence of functionalson bases, Studia Math. 16 (1958), 335–352.[Ost-Mn] A. J. Ostaszewski, Monotone normality <strong>and</strong> G δ -diagonals in the classof inductively generated spaces, Topology, Vol. II (Proc. Fourth Colloq.,Budapest, 1978), pp. 905–930, Colloq. Math. Soc. János Bolyai, 23,North-Holl<strong>and</strong>, 1980.[Ost-knit] A. J. Ostaszewski, Regular variation, <strong>topological</strong> dynamics, <strong>and</strong> the UniformBoundedness Theorem, Topology Proc., 36 (2010), 305-336.[Ost-AH] A. J. Ostaszewski, Analytically heavy spaces: Analytic Baire <strong>and</strong> analyticCantor theorems, preprint.[Ost-LB3] A. J. Ostaszewski, Beyond Lebesgue <strong>and</strong> Baire: III. Analyticity <strong>and</strong> shiftcompactness,preprint.[Ost-Joint] A. J. Ostaszewski, Continuity in <strong>groups</strong>: one-sided to joint, preprint.[Ost-AB] A. J. Ostaszewski, Analytic Baire spaces, preprint.[Oxt1] J. C. Oxtoby, Cartesian products of Baire spaces, Fund. Math. 49 (1960),157-166.[Oxt2] J. C. Oxtoby, Measure <strong>and</strong> category, 2nd ed., Grad. Texts Math. 2,Springer, New York, 1980.[Par]K. R. Parthasarathy, Probability measures on metric spaces. AcademicPress, New York, 1967 (reprinted AMS, 2005).[PRV] K. R. Parthasarathy, R. Ranga Rao, S.R.S Varadhan, Probability distributionson locally compact abelian <strong>groups</strong>, Illinois J. Math. 7 (1963),337–369.[Pav] O. Pavlov, A Lindelöf <strong>topological</strong> group whose square is not normal,preprint.[PeSp] J. Peetre <strong>and</strong> G. Sparr, Interpolation of normed abelian <strong>groups</strong>. Ann.Mat. Pura Appl. (4) 92 (1972), 217–262.[Pes]V. Pestov, Review of [Ne], MathSciNet MR1473172 (98i:22003).[Pet1] B. J. Pettis, On continuity <strong>and</strong> openness of homomorphisms in <strong>topological</strong><strong>groups</strong>, Ann. of Math. (2) 52.2 (1950), 293–308.[Pet2] B. J. Pettis, Remarks on a theorem of E. J. McShane, Proc. Amer. Math.Soc. 2.1 (1951), 166–171.[Pet3] B. J. Pettis, Comments on open homomorphisms, Proc. Amer. Math.Soc. 8.1 (1957), 583–586.
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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
- 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: Normed groups 147[Kak] S. Kakutani,
- Page 155: Normed groups 151[So]R. M. Solovay,