124 N. H. Bingham <strong>and</strong> A. J. OstaszewskiConsider x with ‖x‖ < δ N (η). Then κ N ‖f(x)‖ ≤ ‖f(x) N ‖ = ‖f(x N )‖ < M. So for xwith ‖x‖ < δ N (η) we have‖f(x)‖ < M/κ N < ε,proving continuity at e.Compare [HJ, Th 2.4.9 p. 382]. The Main Theorem of [BOst-Thin] may be given acombinatorial restatement in the group setting. We need some further definitions.Definition. For G a metric group, let C(G) = C(N, G) := {x ∈ G N : x is convergent}denote the sequence space of G. For x ∈ C(G) we writeWe make C(G) into a group by settingL(x) = lim n x n .x · y : = 〈x n y n : n ∈ N〉.Thus e = 〈e G 〉 <strong>and</strong> x −1 = 〈x −1n 〉. We identify G with the subgroup of constant sequences,that isT = {〈g : n ∈ N〉 : g ∈ G}.The natural action of G or T on C(G) is then tx := 〈tx n : n ∈ N〉. Thus 〈g〉 = ge, <strong>and</strong>then tx = te · x.Definition. For G a group, a set G of convergent sequences u = 〈u n : n ∈ N〉 in c(G) isa G-ideal in the sequence space C(G) if it is a subgroup closed under the mutiplicativeaction of G, <strong>and</strong> will be termed complete if it is closed under subsequence formation.That is, a complete G-ideal in C(G) satisfies(i) u ∈ G implies tu = 〈tu n 〉 ∈ G, for each t in G,(ii) u, v ∈ G implies that uv −1 ∈ G,(iii) u ∈ G implies that u M := {u m : m ∈ M} ∈ G for every infinite M.If G satisfies (i) <strong>and</strong> u, v ∈ G implies only that uv ∈ G, we say that G is a G-subidealin C(G).Remarks. 0. In the notation of (iii) above, if G is merely an ideal then G ∗ = {u M : foru ∈ t <strong>and</strong> M ⊂ N} is a complete G-ideal; indeed tu M = (tu) M<strong>and</strong> u M v −1M= (uv−1 ) M<strong>and</strong> u MM ′ = u M ′ for M ′ ⊂ M.1. We speak of a Euclidean sequential structure when G is the vector space R d regardedas an additive group.2. The conditions (i) <strong>and</strong> (ii) assert that G is similar in structure to a left-ideal, beingclosed under multiplication by G <strong>and</strong> a subgroup of C(G).3. We refer only to the combinatorial properties of C(G); but one may give C(G) a pseudonormby setting‖x‖ c := d G (Lx, e) = ‖Lx‖, where Lx := lim x n .
<strong>Normed</strong> <strong>groups</strong> 125The corresponding pseudo-metric isd(x, y) := lim d G (x n , y n ) = d G (Lx, Ly).We may take equivalence of sequences with identical limit; then C(G) ∼ becomes a normedgroup (cf. Th. 3.38). However, in our theorem below we do not wish to refer to such anequivalence.Definitions. For a family F of functions from G to H, we denote by F(T ) the family{f|T : f ∈ F} of functions in F restricted to T ⊆ G. Let us denote a convergent sequencewith limit x 0 , by {x n } → x 0 . We say the property Q of functions (property being regardedset-theoretically, i.e. as a family of functions from G to H) is sequential on T iff ∈ Q iff (∀{x n : n > 0} ⊆ T )[({x n } → x 0 ) =⇒ f|{x n : n > 0} ∈ Q({x n : n > 0})].If we further require the limit point to be enumerated in the sequence, we call Q completelysequential on T iff ∈ Q iff (∀{x n } ⊆ T )[({x n } → x 0 ) =⇒ f|{x n } ∈ Q({x n })].Our interest rests on properties that are completely sequential; our theorem below containsa condition referring to completely sequential properties, that is, the condition isrequired to hold on convergent sequences with limit included (so on a compact set), ratherthan on arbitrary sequences.Note that if Q is (completely) sequential then f|{x n } ∈ Q({x n }) iff f|{x n : n ∈ M} ∈Q({x n : n ∈ M}), for every infinite M.Definition. Let h : G → H, with G, H metric <strong>groups</strong>. Say that a sequence u = {u n } isQ-good for h ifh|{u n } ∈ Q|{u n },<strong>and</strong> putG hQ = {u : h|{u n } ∈ Q|{u n }}.If Q is completely sequential, then u is Q-good for h iff every subsequence of u is Q-goodfor h, so that G hQ is a G-ideal iff it is a complete G-ideal. One then has:Lemma 11.23. If Q is completely sequential <strong>and</strong> F preserves Q under shift <strong>and</strong> multiplication<strong>and</strong> division on compacts, then G hQ for h ∈ F is a G-ideal.Theorem 11.24 (Analytic Automaticity Theorem - combinatorial form). Suppose thatfunctions of F having Q on G have P on G, where Q is a property of functions from Gto H that is completely sequential on G.Suppose that, for all h ∈ F, G hQ , the family of Q-good sequences is a G-ideal. Then, forany analytic set T spanning G, functions of F having Q on T have P on G.
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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
- 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 and 88: Normed groups 83is continuous at th
- Page 89 and 90: Normed groups 85compact. Evidently,
- 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: Normed groups 123Hence, as t i n
- 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 and 154: Normed groups 149fields. I. Basic p
- Page 155: Normed groups 151[So]R. M. Solovay,