Basic Analysis – Gently Done Topological Vector Spaces

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6: Topological Vector Spaces 59 Proposition 6.17 Let A be a balanced subset of a topological vector space. Then A, the closure of A, is balanced and, if 0 is an interior point of A, the interior of A is balanced. Proof For any t ∈ K with 0 < |t| ≤ 1, we have tA = M t (A) = M t (A) = tA ⊆ A, since M t is a homeomorphism. This clearly also holds for t = 0 and so A is balanced. Suppose that 0 ∈ IntA. Then, as before, for 0 < |t| ≤ 1, tIntA = Int(tA) ⊆ tA ⊆ A. But tIntA is open and so tIntA ⊆ IntA. This is also valid for t = 0 and it follows that IntA is balanced. Proposition 6.18 Any neighbourhood of 0 in a topological vector space contains a balanced neighbourhood of 0. Proof Let V be a given neighbourhood of 0 in a topological vector space X. Since scalar multiplication is continuous (at (0,0)), there is δ > 0 and a neighbourhood U of 0 such that if |β| < δ and x ∈ U, then βx ∈ V, that is, βU ⊆ V if |β| < δ. Put W = � δ |β|
60 Basic Analysis Corollary 6.20 Suppose that X and Y are normed spaces and T : X → Y is a linear map. The following statements are equivalent. (i) T is continuous at 0. (ii) T is continuous. (iii) There is C > 0 such that �Tx� ≤ C�x�, for all x ∈ X. Proof By the proposition, (i) and (ii) are equivalent, and certainly (iii) implies (i). We shall show that (i) implies (iii). Let V be the neighbourhood of 0 in Y given by V = {y ∈ Y : �y� ≤ 1}. The continuity of T at 0 implies that there is a neighbourhood U of 0 in X such that T(U) ⊆ V. But U contains a ball of radius r, for some r > 0, so that �x� < r implies that Tx ∈ V, that is, �Tx� ≤ 1. Hence, for any x ∈ X, with x �= 0, we have that �T(rx/2�x�)� ≤ 1. It follows that �Tx� ≤ 2 �x� x ∈ X, x �= 0, r and this inequality persists for x = 0. Theorem 6.21 Suppose that Λ is a linear functional on a topological vector space X, and suppose that Λ(x) �= 0 for some x ∈ X. The following statements are equivalent. (i) Λ is continuous. (ii) kerΛ is closed. (iii) Λ is bounded on some neighbourhood of 0. Proof Since kerΛ = Λ −1 ({0}) and {0} is closed in K, it is clear that (i) implies (ii). Suppose that kerΛ is closed. By hypothesis, kerΛ �= X, so the complement of kerΛ is a non-empty open set in X. Hence there is a point x /∈ kerΛ and V = X\kerΛ is a (open) neighbourhood of x. Let U = V −x. Then U is an open neighbourhood of 0 and so contains a balanced neighbourhood W, say, of 0. Thus x+W ⊆ V and so (x+W)∩kerΛ = ∅. The linearity of Λ implies that the set Λ(W) is a balanced subset of K and so is either bounded or equal to the whole of K. If Λ(W) = K, there is some w ∈ W such that Λ(w) = −Λ(x). This gives Λ(x + w) = 0, which is impossible since x + W contains no points of the kernel of Λ. It follows that Λ(W) is bounded, that is, there is some K > 0 such that |Λ(w)| < K for all w ∈ W. Hence (ii) implies (iii). If (iii) holds, there is some neighbourhood V of 0 and some K > 0 such that V is a neighbourhood K V we have ε x ∈ V so that |Λ(K ε x)| < K, that is, |Λ(x)| < |Λ(v)| < K for all v ∈ V. Let ε > 0 be given. Then ε K of 0, and if x ∈ ε K ε. It follows that Λ is continuous at 0, and hence Λ is continuous on X, by Proposition 6.19. Department of Mathematics King’s College, London
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Basic Analysis - Gently Done Topolo
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Contents 1 Topological Spaces . . .
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2 Basic Analysis Definition 1.3 A t
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4 Basic Analysis Proof The statemen
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6 Basic Analysis Proposition 1.20 L
Page 11 and 12: 8 Basic Analysis Theorem 1.27 Suppo
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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 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 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: 58 Basic Analysis Proposition 6.13
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
Page 75 and 76: 72 Basic Analysis at 0 is to say th
Page 77 and 78: 74 Basic Analysis For the last part
Page 79 and 80: 76 Basic Analysis are equivalent; s
Page 81 and 82: 78 Basic Analysis C = {f ∈ X : p
Page 83 and 84: 80 Basic Analysis Hence, for x ∈
Page 85 and 86: 82 Basic Analysis Thus, for t ∈ K
Page 87 and 88: 8. Banach Spaces In this chapter, w
Page 89 and 90: 86 Basic Analysis 5. The Banach spa
Page 91 and 92: 88 Basic Analysis We shall apply th
Page 93 and 94: 90 Basic Analysis Proposition 8.7 F
Page 97 and 98: 94 Basic Analysis and we deduce tha
Page 99 and 100: 96 Basic Analysis Theorem 8.16 Supp
Page 101 and 102: 98 Basic Analysis for x ∈ X = ℓ
Page 103 and 104: 100 Basic Analysis and so �ϕ x
Page 105 and 106: 102 Basic Analysis The next result
Page 107 and 108: 104 Basic Analysis Now we consider
Page 109 and 110: 106 Basic Analysis Theorem 9.13 For
Page 111 and 112: 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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