26 N. H. Bingham <strong>and</strong> A. J. Ostaszewskishowing that the product of two slowly varying functions is slowly varying, sincef(tx)f(t) −1 → e iff ‖f(tx)f(t) −1 ‖ → 0.3. <strong>Normed</strong> <strong>versus</strong> <strong>topological</strong> <strong>groups</strong>By the Birkhoff-Kakutani Theorem above (Th. 2.19) any metrizable <strong>topological</strong> grouphas a right-invariant equivalent metric, <strong>and</strong> hence is a normed group. Theorem 3.4 belowestablishes a converse: a normed group is a <strong>topological</strong> group provided all its shifts(both right <strong>and</strong> left-sided) are continuous, i.e. provided the normed group is semi<strong>topological</strong>(see [ArRez]). This is not altogether surprising, in the light of known results onsemi<strong>topological</strong> <strong>groups</strong>: assuming that a group T is metrizable, non-meagre <strong>and</strong> analyticin the metric, <strong>and</strong> that both left <strong>and</strong> right-shifts are continuous, then T is a <strong>topological</strong>group (see e.g. [THJ] for several results of this kind in [Rog2, p. 352]; compare also[Ell2] <strong>and</strong> the literature cited under Remarks 2 in Section 2). The results here are cognate,<strong>and</strong> new because a normed group has a one-sided rather than a two-sided topology.We will also establish the equivalent condition that all conjugacies γ g (x) := gxg −1 arecontinuous; this has the advantage of being stated in terms of the norm, rather than interms of one of the associated metrics. As inner automorphisms are homomorphisms, thiscondition ties the structure of normed <strong>groups</strong> to issues of automatic continuity of homomorphisms:automatic continuity forces a normed group to be a <strong>topological</strong> group (<strong>and</strong>the homomorphisms to be homeomorphisms). <strong>Normed</strong> <strong>groups</strong> are thus either <strong>topological</strong>or pathological, as noted in the Introduction.The current section falls into three parts. In the first we characterize <strong>topological</strong><strong>groups</strong> in the category of normed <strong>groups</strong> <strong>and</strong> so in particular, using norms, characterizealso the Klee <strong>groups</strong> (<strong>topological</strong> <strong>groups</strong> which have an equivalent bi-invariant metric).Then we study continuous automorphisms in relation to Lipschitz norms. In the thirdsubsection we demonstrate that a small amount of regularity forces a normed group tobe a <strong>topological</strong> group.3.1. Left <strong>versus</strong> right-shifts: Equivalence Theorem. As we have seen in Th. 2.3, agroup-norm defines two metrics: the right-invariant metric which we denote as usual byd R (x, y) := ‖xy −1 ‖ <strong>and</strong> the conjugate left-invariant metric, here to be denoted d L (x, y) :=d R (x −1 , y −1 ) = ‖x −1 y‖. There is correspondingly a right <strong>and</strong> left metric topology whichwe term the right or left norm topology. We favour this over ‘right’ or ‘left’ normed <strong>groups</strong>rather than follow the [HS] paradigm of ‘right’ <strong>and</strong> ‘left’ <strong>topological</strong> semi<strong>groups</strong>. We write→ R for convergence under d R etc. Recall that both metrics give rise to the same norm,since d L (x, e) = d R (x −1 , e) = d R (e, x) = ‖x‖, <strong>and</strong> hence define the same balls centeredat the origin e:B d R(e, r) := {x : d(e, x) < r} = B d L(e, r).
<strong>Normed</strong> <strong>groups</strong> 27Denoting this commonly determined set by B(r), we have seen in Proposition 2.5 thatB R (a, r) = {x : x = ya <strong>and</strong> d R (a, x) = d R (e, y) < r} = B(r)a,B L (a, r) = {x : x = ay <strong>and</strong> d L (a, x) = d L (e, y) < r} = aB(r).Thus the open balls are right- or left-shifts of the norm balls at the origin. This is bestviewed in the current context as saying that under d R the right-shift ρ a : x → xa is rightuniformly continuous, sinced R (xa, ya) = d R (x, y),<strong>and</strong> likewise that under d L the left-shift λ a : x → ax is left uniformly continuous, sinced L (ax, ay) = d L (x, y).In particular, under d R we have y → R b iff yb −1 → R e, as d R (e, yb −1 ) = d R (y, b). Likewise,under d L we have x → L a iff a −1 x → L e, as d L (e, a −1 x) = d L (x, a).Thus either topology is determined by the neighbourhoods of the identity (origin)<strong>and</strong> according to choice makes the appropriately sided shift continuous; said anotherway, the topology is determined by the neighbourhoods of the identity <strong>and</strong> the chosenshifts. We noted earlier that the triangle inequality implies that multiplication is jointlycontinuous at the identity e, as a mapping from (X, d R ) to (X, d R ). Likewise inversion isalso continuous at the identity by the symmetry axiom. (See Theorem 2.19 ′ .) To obtainsimilar results elsewhere one needs to have continuous conjugation, <strong>and</strong> this is linkedto the equivalence of the two norm topologies (see Th. 3.4). The conjugacy map underg ∈ G (inner automorphism) is defined byγ g (x) := gxg −1 .Recall that the inverse of γ g is given by conjugation under g −1 <strong>and</strong> that γ g is ahomomorphism. Its continuity, as a mapping from (X, d R ) to (X, d R ), is thus determinedby behaviour at the identity, as we verify below. We work with the right topology (underd R ), <strong>and</strong> sometimes leave unsaid equivalent assertions about the isometric case of (X, d L )replacing (X, d R ).Lemma 3.1. The homomorphism γ gright-to-right continuous at e.is right-to-right continuous at any point iff it isProof. This is immediate since x → R a if <strong>and</strong> only if xa −1 → R e <strong>and</strong> γ g (x) → R γ g (a) iffγ g (xa −1 ) → R γ g (e), since‖gxg −1 (gag −1 ) −1 ‖ = ‖gxa −1 g −1 ‖.We note that, by the Generalized Darboux Theorem (Th. 11.22), if γ g is locallynorm-bounded <strong>and</strong> the norm is N-subhomogeneous (i.e. a Darboux norm – there areconstants κ n → ∞ with κ n ‖z‖ ≤ ‖z n ‖), then γ g is continuous. Working under d R , wewill relate inversion to left-shifts. We begin with the following, a formalization of anearlier observation.
- Page 1 and 2: N. H. BINGHAM and A. J. OSTASZEWSKI
- Page 3 and 4: Normed groups 3ContentsContents . .
- Page 5 and 6: 1. IntroductionGroup-norms, which b
- Page 7 and 8: Normed groups 3Topological complete
- Page 9 and 10: Normed groups 5abelian group has se
- Page 11 and 12: Normed groups 74 (Topological permu
- Page 13 and 14: Normed groups 9The following result
- Page 15 and 16: Normed groups 11Corollary 2.4. For
- Page 17 and 18: Normed groups 13More generally, for
- Page 19 and 20: Normed groups 15definitions, our pr
- Page 21 and 22: Normed groups 17so that fg is in th
- Page 23 and 24: Normed groups 19(iii) The ¯d H -to
- Page 25 and 26: Normed groups 21so‖αβ‖ ≤
- Page 27 and 28: Normed groups 23Remark. Note that,
- Page 29: Normed groups 25shows that [z n , y
- Page 33 and 34: 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
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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
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Normed groups 97Proof. Note that‖
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Normed groups 99Taking h(x) := ‖
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Normed groups 1019. The Semigroup T
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Normed groups 103Theorem 9.5 (Semig
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Normed groups 105By the Category Em
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Normed groups 107Proof. Say f is bo
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Normed groups 109Thus G is locally
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Normed groups 111Theorem 10.10 (Bar
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Normed groups 113K-analyticity was
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Normed groups 115Theorem 11.6 (Disc
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Normed groups 117restricted to X\M
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Normed groups 119groups need not be
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Normed groups 121Proof. In the meas
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Normed groups 123Hence, as t i n
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Normed groups 125The corresponding
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Normed groups 127(t, x) ✛✻Φ T
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Normed groups 129Fix s. Since s is
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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,