84 N. H. Bingham <strong>and</strong> A. J. OstaszewskiLikewise, for quasi all t ∈ T there is an infinite set M t such that{tz m : m ∈ M t } ⊆ T.Proof. Apply Th. 6.2, taking for d a right-invariant metric, d X R say; the continuous mapsψ n (t) = z n t satisfy d X R (z nt, t) = ‖z n ‖ H → 0, so converge to the identity. Likewise takingfor d a left-invariant metric d X L say, the continuous maps ψ n(t) = tz n satisfy d X R (tz n, t) =‖z n ‖ H → 0, so again converge to the identity.As a corollary of the KBD Theorem of Section 5 (Th. 5.1) we have the followingimportant result known for <strong>topological</strong> <strong>groups</strong> (see [RR-TG], Rogers [Jay-Rog, p. 48], <strong>and</strong>[Kom1] for the <strong>topological</strong> vector space setting) <strong>and</strong> here proved in the metric setting.Theorem 6.5 (Piccard-Pettis Theorem – Piccard [Pic1], [Pic2], Pettis [Pet1], [RR-TG] cf.[BOst-TRII]). In a normed <strong>topological</strong> group whose norm topology is Baire: for A Baire<strong>and</strong> non-meagre (in the norm topology), the sets AA −1 <strong>and</strong> A −1 A both have non-emptyinterior.Proof. Suppose otherwise. We work first with the right-invariant metric d R (x, y) =‖xy −1 ‖ <strong>and</strong> assume A −1 is Baire non-meagre in the right-norm topology. Consider theset A −1 A. Suppose the conclusion fails for A −1 A, for each integer n = 1, 2, ... there isz n ∈ B 1/n (e)\A −1 A; hence z n → z 0 = e. Applying either the KBD Theorem for <strong>topological</strong>lycomplete normed <strong>groups</strong> or its variant for <strong>topological</strong> <strong>groups</strong>, there is a ∈ A suchthat for infinitely many naz n ∈ A, or z n ∈ A −1 A,a contradiction. Thus, for some n, the open ball B 1/n (e) is contained in A −1 A. We nextconsider the set AA −1 . As the inversion mapping x → x −1 is a homeomorphism (infact an isometry, see Prop. 2.3) from the right- to the left-norm topology, the set A isBaire non-meagre in the left-norm topology iff A −1 is Baire non-meagre in the right-normtoplogy. But the inversion mapping carries the ball B 1/n (e) into itself, <strong>and</strong> so we maynow conclude that AA −1 contains an open ball B 1/n (e), as (A −1 ) −1 = A.One says that a set A is thick if e is an interior point of AA −1 (see e.g. [HJ, Section3.4] ). The next result (proved essentially by the same means) applied to the additivegroup R implies the Kestelman-Borwein-Ditor ([BOst-LBII]) theorem on the line. Thename used here refers to a similar (weaker) property studied in Probability Theory (inthe context of probabilities regarded as a semigroup under convolution, for which see[PRV], or [Par, 3.2 <strong>and</strong> 3.5], [BlHe], [Hey]). We need a definition.Definition (cf.[BOst-StOstr]). In a normed <strong>topological</strong> group G, say that a set A is(properly) right-shift compact, resp. strongly right-shift compact if, for any sequence ofpoints a n in A, (resp. in G) there is a point t <strong>and</strong> a subsequence {a n : n ∈ M t } such thata n t lies entirely in A <strong>and</strong> converges through M t to a point a 0 t in A; similarly for left-shift
<strong>Normed</strong> <strong>groups</strong> 85compact. Evidently, finite Cartesian products of shift-compact sets are shift-compact.Thus a right-shift compact set A is precompact. (If the subsequence a m t converges toa 0 t, for m in M t , then likewise a m converges to a 0 , for m in M t .)Proposition 6.6. In a normed <strong>topological</strong> group, if a subgroup S is locally right-shiftcompact, then S is closed <strong>and</strong> locally compact. Conversely, a closed, locally compact subgroupis locally right-shift compact.Proof. Suppose that a n → a 0 with a n ∈ S. If a m t → a 0 t ∈ S down a subset M, thena 0 t(a m t) −1 = a 0 a −1m ∈ S for m ∈ M. Hence also a 0 = a 0 a −1m a m ∈ S for m ∈ M. Thus Sis closed.Example. In the additive group R, the subgroup Z is closed <strong>and</strong> locally compact, soshift-compact. Of course, Z is too small to contain shifts of arbitrary null sequences. Wereturn to this matter in the remarks after Th. 7.7, where we distinguish between propershift-compactness as here (so that we are concerned only with sequences in a given set)<strong>and</strong> null-shift compactness where we are concerned with shifting subsequences of arbitrarysequences into a given set.Example. Note that A ⊆ R is density-open (open in the density topology) iff each pointof A is a density point of A. Suppose a 0 is a limit point (in the usual topology) of such aset A; then, for any ε > 0, we may find a point α ∈ A within ε/2 of a 0 <strong>and</strong> hence somet ∈ A within ε/2 of the point α such that some subsequence t + a m is included in A, withlimit t + a 0 <strong>and</strong> with |t| < ε. That is, a density-open set is strongly shift-compact.Remark. Suppose that a n = (a i n) ∈ A = ∏ A i . Pick t i <strong>and</strong> inductively infinite M i ⊆M i−1 so that a i nt i → a i 0t i along n ∈ M i with a i nt i ∈ A i for n ∈ ω. Diagonalize M i bysetting M := {m i }, where m n+1 = min{m ∈ M n+1 : m > m n }. Then the subsequence{a m : m ∈ M} satisfies, for each J finite,pr J a m t ⊆ ∏ j∈J A j for eventually all m ∈ M,<strong>and</strong> so in the product topology a m t → a 0 t through M, where (a i )(t i ) is defined to be(a i t i ).Theorem 6.7 (Shift-Compactness Theorem). In a normed <strong>topological</strong> group G, for Aprecompact, Baire <strong>and</strong> non-meagre, the set A is properly right-shift compact, i.e., for anysequence a n ∈ A, there are t ∈ G <strong>and</strong> a ∈ A such that a n t ∈ A <strong>and</strong> a n t → a down asubsequence. Likewise the set A is left-shift compact.
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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
- Page 37 and 38: Normed groups 33(ii) For α ∈ H u
- Page 39 and 40: Normed groups 35Definition. A group
- Page 41 and 42: Normed groups 37We now give an expl
- Page 43 and 44: Normed groups 39Theorem 3.19 (Abeli
- Page 45 and 46: Normed groups 412. Further recall t
- Page 47 and 48: Normed groups 43Theorem 3.22 (Lipsc
- Page 49 and 50: Normed groups 45Proof. Z γ = G (cf
- Page 51 and 52: Normed groups 47Theorem 3.30. Let G
- Page 53 and 54: Normed groups 49Remark. On the matt
- Page 55 and 56: Normed groups 51As for the conclusi
- Page 57 and 58: Normed groups 53By (C-adm), we may
- Page 59 and 60: Normed groups 55equipped with an in
- Page 61 and 62: Normed groups 57Proof. To apply Th.
- Page 63 and 64: Normed groups 59Definition. A point
- Page 65 and 66: Normed groups 61Proposition 3.46 (M
- Page 67 and 68: Normed groups 63Thus ω δ (s) ≤
- Page 69 and 70: Normed groups 65Remark. In the penu
- Page 71 and 72: Normed groups 67The result confirms
- Page 73 and 74: Normed groups 69Proof. By the Baire
- Page 75 and 76: Normed groups 715. Generic Dichotom
- Page 77 and 78: Normed groups 73Returning to the cr
- Page 79 and 80: Normed groups 75Examples. Here are
- Page 81 and 82: Normed groups 77cf. [Eng, 4.3.23].)
- Page 83 and 84: Normed groups 79Remarks. 1. See [Fo
- Page 85 and 86: Normed groups 81Theorem 6.1 (Catego
- Page 87: Normed groups 83is continuous at th
- Page 91 and 92: 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
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Normed groups 135Definition. Let {
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Normed groups 137However, whilst th
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Normed groups 139embeddable, 14enab
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Normed groups 141Bibliography[AL]J.
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Normed groups 143Series 378, 2010.[
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Normed groups 145abelian groups, Ma
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Normed groups 147[Kak] S. Kakutani,
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Normed groups 149fields. I. Basic p
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Normed groups 151[So]R. M. Solovay,