Normed versus topological groups: Dichotomy and duality

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82 N. H. Bingham and A. J. OstaszewskiTheorem 6.2 (First Verification Theorem for weak category convergence). For (X, d) ametric space, if ψ n converges to the identity under ˆd = ˆd H , then ψ n satisfies the weakcategory convergence condition (wcc).Proof. It is more convenient to prove the equivalent statement that ψn−1 satisfies thecategory convergence condition.Put z n = ψ n (z 0 ), so that z n → z 0 . Let k be given.Suppose that y ∈ B ε (z 0 ), i.e. r = d(y, z 0 ) < ε. For some N > k, we have ε n = d(ψ n , id)
Normed groups 83is continuous at the identity, the commutator condition has the equivalent formulationthat (xz n x −1 )zn−1 → e, and this combined with the triangle inequality‖xz n x −1 ‖ = ‖xz n x −1 z −1nz n ‖ ≤ ‖xz n x −1 z −1n‖ + ‖zn−1 ‖implies that γ x (z) is continuous at z = e. As x is arbitrary Theorem 3.4 again impliesthat X is a topological group.Remark. Let X be given the right norm topology and let z n → e. For the homeomorphismψ n (x) = ρ n (x) = xz n one has⋂n≥k B ε(e X )\ψ n (B ε (e)) = ∅, k = 1, 2, ....Nevertheless, we cannot deduce from here that⋂n≥k B ε(x)\ψ n (B ε (x)) = ∅.The obstruction is thatψ n (B ε (x)) = B ε (x)z n = B ε (e)xz n ≠ B ε (e)z n x.A natural argument that fails is to say that for z ∈ B ε (x) with z = yx one has d R (yx, x) =‖y‖ < ε and so for n larged R (yx, z n x) ≤ d R (yx, x) + d R (x, z n x) = ‖y‖ + ‖z n ‖ < ε.But this gives only that z = yx ∈ B ε (z n x) = B ε (e)z n x rather than in z ∈ B ε (e)xz n .Thus one is tempted to finesse this difficulty by requiring additionally that, for fixed x,z n xzn−1 x −1 → e as n → ∞, on the grounds that for large nd R (yx, xz n ) ≤ d R (yx, x) + d R (x, z n x) + d R (z n x, xz n ) < ε.This does indeed yields z = yx ∈ B ε (xz n ) = B ε (x)z n = ψ n (B ε (x)), as desired. However,the assertion (ii) shows that we have appealed to a topological group structure.We mention that the expected modification of the above argument becomes valid underthe ambidextrous topology generated by d S := max{d R , d L }. However, shifts are not thenguaranteed to be continuous.As a first corollary we have the following topological result; we deduce later, also ascorollaries, measure-theoretic versions in Theorems 7.6 and 11.14. Here in the left-sidedcategory variant we refer to the left-shifts ψ n (t) = z n t which converge to the identityunder a right-invariant metric, but, as we also need these shifts to be homeomorphisms(so right-to-right continuous in the sense of Section 3), it is necessary to require thenormed group to be a topological group – by the last Remark. We thus obtain herea weakened result. (Note that ‘normed topological group’ is synonymous with ‘metricgroup’.)Corollary 6.4 (Topological Kestelman-Borwein-Ditor Theorem). In a normed topologicalgroup X let {z n } → e X be a null sequence. If T is a Baire subset of X, then forquasi all t ∈ T there is an infinite set M t such that{z m t : m ∈ M t } ⊆ T.
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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
Page 35 and 36: 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: Normed groups 81Theorem 6.1 (Catego
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 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‖ −
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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,
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Normed versus topological groups: Dichotomy and duality

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