Basic Analysis – Gently Done Topological Vector Spaces

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1: Topological Spaces 11 The local version of a base is often useful, not least for purposes of economy. Definition 1.39 Let (X,T) be a topological space, and, for x ∈ X, let V x denote the collection of all neighbourhoods of x. A subset N x ⊆ V x is said to be a (local) neighbourhood base at x (or a fundamental system of neighbourhoods of x) if any element of V x contains some element of the subcollection N x . A topological space is said to be first countable if each of its points possesses a countable neighbourhood base. It is said to be second countable if it possesses a countable base. (Evidently, a second countable space is first countable—those members of the base containing any given point constitute a countable neighbourhood base for that particular point.) Example 1.40 In any metric space (X,d), the countable collection of disks � {x ′ : d(x,x ′ ) < 1 n }� is a neighbourhood base at the point x ∈ X. Consequently, n∈N every metrizable topological space is first countable. The space Rn , equipped with the usual (metric) topology, is an example of a second countable space. Indeed, a countable base is given by the collection of open balls with rational radii and centres with rational coordinates. Proposition 1.41 Suppose that (X,T) and (Y,S) are topological spaces. A map f : X → Y is continuous if and only if for each x ∈ X and local neighbourhood bases N x and N f(x) at x and f(x), respectively, it is true that for any V ∈ N f(x) there is U ∈ N x such that f(U) ⊆ V. Proof Suppose that f is continuous. Let x ∈ X and let N x and N f(x) be neighbourhood bases at x and f(x), respectively. Let V ∈ N f(x) . Then there is an open set G such that f(x) ∈ G ⊆ V. By continuity, f −1 (G) is an open neighbourhood of x and so there is some U ∈ N x such that x ∈ U ⊆ f −1 (G). Thus f(U) ⊆ V. Conversely, suppose that for any x ∈ X and local neighbourhood bases N x and N f(x) and for any V ∈ N f(x) , there is some U ∈ N x such that f(U) ⊆ V. We wish to show that f is continuous. Let G be an open set in Y. If f −1 (G) = ∅, there is no more to prove, so suppose that x ∈ f −1 (G). Then f(x) ∈ G and there is some V in N f(x) such that V ⊆ G. But then, by hypothesis, there is some U ∈ N x such f(U) ⊆ V. Let W be any open neighbourhood of x with W ⊆ U. Then f(W) ⊆ V and so certainly it is true that W ⊆ f −1 (G). Hence every point of f −1 (G) is an interior point and we conclude that f −1 (G) is open and so f is continuous. The following is a generalisation of the standard ε-δ definition of the continuity at a given point of a real function of a real variable to the present context. Definition 1.42 Let (X,T) and (Y,S) be topological spaces and suppose that f : X → Y is a given map. We say that f is continuous at the point x ∈ X if for any neighbourhood V of f(x) in Y there is a neighbourhood U of x in X such that U ⊆ f −1 (V), i.e., f(U) ⊆ V.
12 Basic Analysis Thepreviouspropositionthenstatesthatamappingfromonetopologicalspace into another is continuous if and only if it is continuous at every point. Department of Mathematics King’s College, London
Page 1 and 2: Basic Analysis - Gently Done Topolo
Page 3 and 4: Contents 1 Topological Spaces . . .
Page 5 and 6: 2 Basic Analysis Definition 1.3 A t
Page 7 and 8: 4 Basic Analysis Proof The statemen
Page 9 and 10: 6 Basic Analysis Proposition 1.20 L
Page 11 and 12: 8 Basic Analysis Theorem 1.27 Suppo
Page 13: 10 Basic Analysis Proposition 1.36
Page 17 and 18: 14 Basic Analysis ample, the proble
Page 19 and 20: 16 Basic Analysis Proposition 2.8 L
Page 21 and 22: 18 Basic Analysis Theorem 2.13 Let
Page 23 and 24: 20 Basic Analysis A point z is a cl
Page 25 and 26: 22 Basic Analysis family {U x : x
Page 27 and 28: 24 Basic Analysis and so � A∩B
Page 29 and 30: 26 Basic Analysis Remark 3.2 Let G
Page 31 and 32: 28 Basic Analysis Proposition 3.7 S
Page 33 and 34: 30 Basic Analysis Proof (version 2)
Page 35 and 36: 4. Separation The Hausdorff propert
Page 37 and 38: 34 Basic Analysis Next we show that
Page 39 and 40: 36 Basic Analysis Q x contains no r
Page 41 and 42: 38 Basic Analysis that g 0 (x) =
Page 43 and 44: 5. Vector Spaces We shall collect t
Page 45 and 46: 42 Basic Analysis Zorn’s lemma ca
Page 47 and 48: 44 Basic Analysis Finally, suppose
Page 49 and 50: 46 Basic Analysis Proposition 5.14
Page 51 and 52: 48 Basic Analysis Nextweconsiderthe
Page 53 and 54: 50 Basic Analysis and so, multiplyi
Page 55 and 56: 52 Basic Analysis so that Λ ↾ M
Page 57 and 58: 54 Basic Analysis subsets of X. We
Page 59 and 60: 56 Basic Analysis Remark 6.7 The co
Page 61 and 62: 58 Basic Analysis Proposition 6.13
Page 63 and 64: 60 Basic Analysis Corollary 6.20 Su
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62 Basic Analysis For each 1 ≤ i
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64 Basic Analysis Corollary 6.29 Le
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7. Locally Convex Topological Vecto
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68 Basic Analysis To show that each
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70 Basic Analysis vector topology o
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72 Basic Analysis at 0 is to say th
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74 Basic Analysis For the last part
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76 Basic Analysis are equivalent; s
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78 Basic Analysis C = {f ∈ X : p
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80 Basic Analysis Hence, for x ∈
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82 Basic Analysis Thus, for t ∈ K
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8. Banach Spaces In this chapter, w
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86 Basic Analysis 5. The Banach spa
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88 Basic Analysis We shall apply th
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90 Basic Analysis Proposition 8.7 F
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92 Basic Analysis Proposition 8.10
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94 Basic Analysis and we deduce tha
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96 Basic Analysis Theorem 8.16 Supp
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98 Basic Analysis for x ∈ X = ℓ
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100 Basic Analysis and so �ϕ x
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102 Basic Analysis The next result
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104 Basic Analysis Now we consider
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106 Basic Analysis Theorem 9.13 For
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108 Basic Analysis Definition 9.19
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110 Basic Analysis According to the
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10. Fréchet Spaces Inthischapter w
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114 Basic Analysis and this is sepa
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116 Basic Analysis For any m,n > N,
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118 Basic Analysis Theorem 10.18 (B
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120 Basic Analysis because X = �
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122 Basic Analysis Corollary 10.23
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124 Basic Analysis Remark 10.27 It
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126 Basic Analysis Define P : X →
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Basic Analysis – Gently Done Topological Vector Spaces

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