Basic Analysis – Gently Done Topological Vector Spaces

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2: Nets 21 Theorem 2.21 A point x in a topological space X is a cluster point of the net (x α ) I if and only if some subnet converges to x. Proof Suppose that x is a cluster point of the net (x α ) I and let N be the family of neighbourhoods of x. Then if A,B ∈ N, we have A∩B ∈ N, and also (x α ) is frequently in each member of N. By the preceding proposition, there is a subnet (y β ) J eventually in each member of N, that is, the subnet (y β ) converges to x. Conversely, suppose that (y β ) β∈J = (x F(β) ) β∈J is a subnet of (x α ) I converging tox. Wemust showthatxisaclusterpointof(x α ) I . Let N beanyneighbourhood of x. Then there is β 0 ∈ J such that x F(β) ∈ N whenever β � β 0 . Since F is cofinal, for any given α ′ ∈ I there is β ′ ∈ J such that F(β) � α ′ whenever β � β ′ . Let β � β 0 and β � β ′ . Then F(β) � α ′ and y β = x F(β) ∈ N. Hence (x α ) I is frequently in N and we conclude that x is a cluster point of the net (x α ) I , as claimed. In a metric space, compactness is equivalent to sequential compactness—the Bolzano-Weierstrass property. In a general topological space, this need no longer be the case. However, there is an analogue in terms of nets. Theorem 2.22 A topological space (X,T) is compact if and only if every net in X has a convergent subnet. Proof Suppose that every net has a convergent subnet. Let {G α } I be an open cover of X with no finite subcover. Let F be the collection of finite subfamilies of theopencover, orderedbyset-theoreticinclusion. ForeachF = {Gα1 ,...,G } ∈ αm F, let xF be any point in X such that xF /∈ �m j=1Gαj . Note that such xF exists since {Gα } has no finite subcover. By hypothesis, the net (xF ) F∈F has a convergent subnet or, equivalently, by the previous theorem, a cluster point x, say. Now, since {Gα } is a cover of X, there is some α ′ such that x ∈ Gα ′. But then, by definition of cluster point, (xF ) F is frequently in Gα ′. Thus, for any F ′ ∈ F, there isF � F ′ ∈ F such that xF ∈ Gα ′. Inparticular, if wetakeF ′ = {Gα ′}, wededuce that there is F = {Gα1 ,...,G αk } such that F � {Gα ′}, that is, {Gα ′} ⊆ F, and such that xF ∈ Gα ′. Hence Gα ′ = G for some 1 ≤ i ≤ k, and αi x F ∈ G αi ⊆ k� Gαj . But x F /∈ � k j=1 G αj , by construction. This contradiction implies that every open cover has a finite subcover, and so (X,T) is compact. For the converse, suppose that (X,T) is compact and let (x α ) I be a net in X. Suppose that (x α ) I has no cluster points. Then, for any x ∈ X, there is an open neighbourhood U x of x and α x ∈ I such that x α /∈ U x whenever α � α x . The j=1
22 <strong>Basic</strong> <strong>Analysis</strong> family {U x : x ∈ X} is an open cover of X and so there exists x 1 ,...,x n ∈ X such that �n i=1Uxi = X. Since I is directed there is α � αi for each i = 1,...,n. But then xα /∈ Uxi for all i = 1,...,n, which is impossible since the U ’s cover X. We xi conclude that (x α ) I has a cluster point, or, equivalently, a convergent subnet. Definition 2.23 A universal net in a topological space (X,T) is a net with the property that, for any subset A of X, it is either eventually in A or eventually in X \A, the complement of A. The concept of a universal net leads to substantial simplification of the proofs of various results, as we will see. Proposition 2.24 If a universal net has a cluster point, then it converges (to the cluster point). In particular, a universal net in a Hausdorff space can have at most one cluster point. Proof Suppose that x is a cluster point of the universal net (x α ) I . Then for each neighbourhood N of x, (x α ) is frequently in N. However, (x α ) is either eventually in N or eventually in X \ N. Evidently, the former must be the case and we conclude that (x α ) converges to x. The last part follows because in a Hausdorff space a net can converge to at most one point. At this point, it is not at all clear that universal nets exist! Examples 2.25 1. It is clear that any eventually constant net is a universal net. In particular, any net with finite index set is a universal net. Indeed, if (x α ) I is a net in X with finite index set I, then I has a maximum element, α ′ , say. The net is therefore eventually equal to x α ′. For any subset A ⊆ X, we have that (x α ) I is eventually in A or eventually in X \A depending on whether x α ′ belongs to A or not. 2. No sequence can be a universal net, unless it is eventually constant. To see this, suppose that (x n ) n∈N is a sequence which is not eventually constant. Then the set S = {x n : n ∈ N} is an infinite set. Let A be any infinite subset of S such that S \A also infinite. Then (x n ) cannot be eventually in either of A or its complement. That is, the sequence (x n ) N cannot be universal. Weshallshowthateverynethasauniversalsubnet. Firstweneedthefollowing lemma. 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 and 14: 10 Basic Analysis Proposition 1.36
Page 15 and 16: 12 Basic Analysis Thepreviouspropos
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: 20 Basic Analysis A point z is a cl
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 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 63 and 64: 60 Basic Analysis Corollary 6.20 Su
Page 65 and 66: 62 Basic Analysis For each 1 ≤ i
Page 67 and 68: 64 Basic Analysis Corollary 6.29 Le
Page 69 and 70: 7. Locally Convex Topological Vecto
Page 71 and 72: 68 Basic Analysis To show that each
Page 73 and 74: 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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