Tao_T.-Analysis_I_(Volume_1)__-Hindustan_Book_Agency(2006)

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Contents <strong>Volume</strong> 1 Preface 1 Introduction 1.1 What is analysis? . 1.2 Why do analysis? . 2 The natural numbers 2.1 The Peano axioms 2.2 Addition . . . . 2.3 Multiplication . 3 Set theory 3.1 Fundamentals ........ . 3.2 Russell's paradox (Optional) 3.3 Functions· ......... . 3.4 Images and inverse images . 3.5 Cartesian products 3.6 Cardinality of sets 4 Integers and rationals 4.1 The integers . . . . . 4.2 The rationals . . . . 4.3 Absolute value and exponentiation xiii 1 1 3 14 16 27 33 37 37 52 55 64 70 76 84 84 92 98
viii 4.4 Gaps in the rational numbers 5 The real numbers 5.1 Cauchy sequences ........... . 5.2 Equivalent Cauchy sequences . . . . . 5.3 The construction of the real numbers . 5.4 Ordering the reals ....... . 5.5 The least upper bound property 5.6 Real exponentiation, part I 6 Limits of sequences 6.1 Convergence and limit laws 6.2 The extended real number system 6.3 Suprema and infima of sequences 6.4 Limsup, liminf, and limit points . 6.5 Some standard limits .... 6.6 Subsequences . . . . . . . . 6. 7 Real exponentiation, part II 7 Series 7.1 Finite series .......... . 7.2 Infinite series ......... . 7.3 Sums of non-negative numbers 7.4 Rearrangement of series 7.5 The root and ratio tests 8 Infinite sets 8.1 Countability. 8.2 Summation on infinite sets. 8.3 Uncountable sets .. 8.4 The axiom of choice 8.5 Ordered sets ..... 9 Continuous functions on R 9.1 Subsets of the real line .. 9.2 The algebra of real-valued functions 9.3 Limiting values of functions . . . . . CONTENTS 103 107 109 114 117 127 133 139 145 145 153 157 160 170 17l 175 179 179 189 195 200 204 208 208 216 224 227 232 242 243 250 253
Page 4: ll:JQ1@ HINDUS TAN Ul!!.rU BOOK AGE
Page 7: To my parents, for everything
Page 12 and 13: CONTENTS 15 Power series 15.1 Forma
Page 14 and 15: Preface This text originated from t
Page 16 and 17: Preface XV more advanced facts abou
Page 18 and 19: Preface xvii tive; much of the key
Page 21 and 22: 2 1. Introduction 1. What is a real
Page 24 and 25: 1.2. Why do analysis? 5 At this poi
Page 26: 1.2. Why do analysis? 7 However, de
Page 33 and 34: Chapter 2 Starting at the beginning
Page 36 and 37: 2.1. The Pean? axioms However, we s
Page 39: 20 2. The natuml numbers Now we can
Page 42 and 43: 2.1. The Peano axioms 23 from appea
Page 45 and 46: 26 2. The natural numbers understoo
Page 48 and 49: 2.2. Addition 29 m)++· But by defi
Page 52 and 53: 2.3. Multiplication 33 Exercise 2.2
Page 55 and 56: 36 2. The natural numbers Proof. Se
Page 57 and 58: 38 3. Set theory that some of the a
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40 3. Set theory A.7). Because of t
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3.1. Fundamentals 43 proof We prove
Page 68:
3.1. Fundamentals We have now accum
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54 3. Set theory Instead, we shall
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3.3. Functions 57 One common way to
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60 3. Set theory Proof. Since g o h
Page 86 and 87:
9.4. Images and inverse images 67 N
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70 3. Set theory This should be com
Page 104 and 105:
4 .1. The integers 85 tage of our f
Page 106 and 107:
4.1. The integers 87 Lemma 4.1.3 (A
Page 108 and 109:
4.1. The integers 89 Jfn is a posit
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92 4. Integers and rationals Proof.
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96 4. Integers and rationals xx-1 =
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4-4· Gaps in the rational numbers
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108 5. The real numbers or more of
Page 156:
5.5. The least upper bound property
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5.6. Real exponentiation, part I 14
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164 6. Limits of sequences Example
Page 228 and 229:
8.1. Countability 209 Jtelllark 8.1
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216 8. Infinite sets Exercise 8.1.4
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8.5. Ordered sets 241 yet another c
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9.2. The algebra of real-valued fun
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g, 1. The intermediate value theore
Page 307 and 308:
Chapter 10 Differentiation of funct
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292 10. Differentiation of function
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300 10. Differentiation of function
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308 11. The Riemann integral (a) X
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318 11. The Riemann integral Proof.
Page 341 and 342:
322 11. The Riemann integral Proof.
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324 11. The Riemann integral for ev
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11.5. Riemann integrability of cont
Page 368 and 369:
Chapter A Appendix: the basics of m
Page 370 and 371:
A.1. Mathematical statements 351 :E
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A.2. Implication 357 Sometimes it i
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364 A. Appendix: the basics of math
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;1.4. Variables and quantifiers 367
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A.4· Variables and quantifiers 371
Page 393 and 394:
374 A. Appendix: the basics of math
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A. 7. Equality 377 takes the value
Page 398 and 399:
A. 7. Equality 379 sin(y) = sin(z2)
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B.l. The decimal representation of
Page 408 and 409:
Index ++ (increment), 18, 56 on int
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INDEX C, C 0 , C 1 , C 2 , Ck, 556
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INDEX of functions, 75, 425 discont
Page 415 and 416:
VIII equivalence of in finite dimen
Page 417 and 418:
X negative: see negation, positive
Page 420 and 421:
INDEX sup norm: see supremum as nor
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Tao_T.-Analysis_I_(Volume_1)__-Hindustan_Book_Agency(2006)

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