Basic Analysis – Gently Done Topological Vector Spaces

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Preface These notes form an enhanced version of the lecture notes for the M. Sc. course, Basic Analysis, presented in the Department of Mathematics at King’s College, London. It is a pleasure to thank Dr. Frank Jellett for many enjoyable discussions. The following books might serve as a good starting point for those wishing to find out more. (For example, Wilansky’s book includes a set of 33 tables of theorems and counterexamples, as well as 1500 problems.) R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. 1, Academic Press, 1983, and Vol. 3, Birkhäuser Boston, 1990 J. L. Kelley, General Topology, D. Van Nostrand, Princeton, New Jersey, 1955; reprinted, Springer-Verlag, New York, 1975 I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, 2nd edition, 1988 J. R. Munkres, Topology—A First Course, Prentice Hall, 1975 M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1 Functional Analysis, Academic Press, revised and enlarged edition, 1980 A. P. Robertson and W. Robertson, Topological Vector Spaces, Cambridge University Press, 1964 W. Rudin, Functional Analysis, McGraw-Hill, 2nd edition, 1973 G. F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, 1963 A. E.Taylorand D. C. Lay,Introduction to Functional Analysis,Wiley,New York, 1980 A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw-Hill, 1978 i Ivan F. Wilde
Contents 1 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 53 7 Locally Convex Topological Vector Spaces . . . . . . . . . . . . . . . 66 8 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 9 The Dual Space of a Normed Space . . . . . . . . . . . . . . . . . . 99 10 Fréchet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 ii
Page 1: Basic Analysis - Gently Done Topolo
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 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
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50 Basic Analysis and so, multiplyi
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52 Basic Analysis so that Λ ↾ M
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54 Basic Analysis subsets of X. We
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56 Basic Analysis Remark 6.7 The co
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58 Basic Analysis Proposition 6.13
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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
Page 75 and 76:
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
Page 103 and 104:
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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