146 N. H. Bingham <strong>and</strong> A. J. Ostaszewski[Han-92] R. W. Hansell, Descriptive Topology, in H. Hušek <strong>and</strong> J. van Mill, RecentProgress in General Topology, 275-315, Elsevier, 1992.[H]J. Harding, Decompositions in Quantum Logic, Trans. Amer. Math. Soc.348 (1996), 1839-1862.[HM] S. Hartman <strong>and</strong> J. Mycielski, On the imbedding of <strong>topological</strong> <strong>groups</strong>into connected <strong>topological</strong> <strong>groups</strong>. Colloq. Math. 5 (1958), 167-169.[HMc] R. C. Haworth <strong>and</strong> R. A. McCoy, Baire spaces, Dissertationes Math. 141(1977), 1-73.[HePo] R.W. Heath, T. Poerio, Topological <strong>groups</strong> <strong>and</strong> semi-<strong>groups</strong> on the realswith the density topology, Conference papers, Oxford 2006 (Conferencein honour of Peter Collins <strong>and</strong> Mike Reed).[Hen] R. Henstock, Difference-sets <strong>and</strong> the Banach-Steinhaus Theorem, Proc.London Math. Soc. (3) 13 (1963), 305-321.[Hey] H. Heyer, Probability measures on locally compact <strong>groups</strong>, Ergebnisseder Mathematik und ihrer Grenzgebiete 94, Springer, 1977.[H-P] E. Hille, <strong>and</strong> R.S. Phillips, Functional analysis <strong>and</strong> semi-<strong>groups</strong>, rev.ed. American Mathematical Society Colloquium Publications, vol. 31.American Mathematical Society, Providence, R. I., 1957.[HS]N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification.Theory<strong>and</strong> applications. de Gruyter Expositions in Mathematics,27. Walter de Gruyter & Co., Berlin, 1998.[HJ]J. Hoffmann-Jørgensen, Automatic continuity, Section 3 of [THJ].[HR]E. Hewitt, K.A. Ross, Abstract harmonic analysis, Vol. I. Structure of<strong>topological</strong> <strong>groups</strong>, integration theory, group representations. Second edition.Grundlehren Wiss., 115. Springer, 1979.[IT1] A. Ionescu Tulcea, <strong>and</strong> C. Ionescu Tulcea, On the lifting property. I,JMAA 3 (1961), 537-546.[IT]A. Ionescu Tulcea, <strong>and</strong> C. Ionescu Tulcea, Topics in the theory of lifting.Ergebnisse Math. 48, Springer, 1969.[Itz]G.L. Itzkowitz, A characterization of a class of uniform spaces that admitan invariant integral, Pacific J. Math. 41 (1972), 123–141.[J]N. Jacobson, Lectures in Abstract Algebra. Vol. I. Basic Concepts, VanNostr<strong>and</strong>, 1951.[Jam] R. C. James, Linearly arc-wise connected <strong>topological</strong> Abelian <strong>groups</strong>,Ann. of Math. (2) 44, (1943). 93–102.[JMW] R. C. James, A.D. Michal, M. Wyman, Topological Abelian <strong>groups</strong> withordered norms, Bull. Amer. Math. Soc. 53, (1947). 770–774.[Jay-Rog] J. Jayne <strong>and</strong> C. A. Rogers, Analytic sets, Part 1 of [Rog2].[Jones1] F. B. Jones, Connected <strong>and</strong> disconnected plane sets <strong>and</strong> the functionalequation f(x + y) = f(x) + f(y), Bull. Amer. Math. Soc. 48 (1942)115-120.[Jones2] F. B. Jones, Measure <strong>and</strong> other properties of a Hamel basis, Bull. Amer.Math. Soc. 48, (1942). 472–481.
<strong>Normed</strong> <strong>groups</strong> 147[Kak] S. Kakutani, Über die Metrisation der topologischen Gruppen, Proc.Imp. Acad. Tokyo 12 (1936) 82-84 (also in Selected Papers, Vol. 1, ed.Robert R. Kallman, Birkhäuser, 1986, 60-62).[Kech] A. S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics156, Springer, 1995.[Kel] J. L. Kelley, General Topology, Van Nostr<strong>and</strong>, 1955.[Kem] J. H. B. Kemperman, A general functional equation, Trans. Amer. Math.Soc. 86 (1957), 28–56.[Kes] H. Kestelman, The convergent sequences belonging to a set, J. LondonMath. Soc. 22 (1947), 130-136.[Klee] V. L. Klee, Invariant metrics in <strong>groups</strong> (solution of a problem of Banach),Proc. Amer. Math. Soc. 3 (1952), 484-487.[Kod] K. Kodaira, Über die Beziehung zwischen den Massen und den Topologienin einer Gruppe, Proc. Phys.-Math. Soc. Japan (3) 23, (1941). 67–119.[Kol] A. Kolmogorov, Zur Normierbarkeit eines allgemeinen topologischen linearenRaumes, Studia Math. 5 (1934), 29-33.[Kom1] Z. Kominek, On the sum <strong>and</strong> difference of two sets in <strong>topological</strong> vectorspaces, Fund. Math. 71 (1971), no. 2, 165–169.[Kom2] Z. Kominek, On the continuity of Q-convex <strong>and</strong> additive functions, Aeq.Math. 23 (1981), 146-150.[Kucz] M. Kuczma, An introduction to the theory of functional equations <strong>and</strong>inequalities. Cauchy’s functional equation <strong>and</strong> Jensen’s inequality, PWN,Warsaw, 1985.[KuVa] K. Kunen <strong>and</strong> J. E. Vaughan, H<strong>and</strong>book of set-theoretic topology, North-Holl<strong>and</strong> Publishing Co., Amsterdam, 1984.[Kur-A] C. Kuratowski, Sur les fonctions représentables analytiquement et lesensembles de première catégorie, Fund. Math. 5 (1924), 75-86.[Kur-B] C. Kuratowski, Sur la propriété de Baire dans les groupes métriques,Studia Math. 4 (1933), 38-40.[Kur-1] K. Kuratowski, Topology, Vol. I., PWN, Warsaw 1966.[Kur-2] K. Kuratowski, Topology, Vol. II., PWN, Warsaw 1968.[Lev] F. Levin, Solutions of equations over <strong>groups</strong>, Bull. Amer. Math. Soc. 68(1962), 603–604.[Levi] S. Levi, On Baire cosmic spaces, General topology <strong>and</strong> its relations tomodern analysis <strong>and</strong> algebra, V (Prague, 1981), 450–454, Sigma Ser.Pure Math., 3, Heldermann, Berlin, 1983.[Low] R. Lowen, Approach spaces. The missing link in the topology-uniformitymetrictriad, Oxford Mathematical Monographs, Oxford UniversityPress, 1997.[Loy] Loy, Richard J. Multilinear mappings <strong>and</strong> Banach algebras. J. LondonMath. Soc. (2) 14 (1976), no. 3, 423–429.[LMZ] J. Lukeš, J. Malý, L. Zajíček, Fine topology methods in real analysis <strong>and</strong>
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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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- 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: Normed groups 145abelian groups, Ma
- Page 153 and 154: Normed groups 149fields. I. Basic p
- Page 155: Normed groups 151[So]R. M. Solovay,