Table Of ContentTexts and Readings in Mathematics 37
Terence Tao
Analysis I
Third Edition
Texts and Readings in Mathematics
Volume 37
Advisory Editor
C.S. Seshadri, Chennai Mathematical Institute, Chennai
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi
Editor
Manindra Agrawal, Indian Institute of Technology Kanpur, Kanpur
V. Balaji, Chennai Mathematical Institute, Chennai
R.B. Bapat, Indian Statistical Institute, New Delhi
V.S. Borkar, Indian Institute of Technology Bombay, Mumbai
T.R. Ramadas, Chennai Mathematical Institute, Chennai
V. Srinivas, Tata Institute of Fundamental Research, Mumbai
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Terence Tao
Analysis I
Third Edition
123
Terence Tao
Department ofMathematics
University of California, LosAngeles
LosAngeles, CA
USA
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To my parents, for everything
Contents
Preface to the second and third editions xi
Preface to the first edition xiii
About the Author xix
1 Introduction 1
1.1 What is analysis? . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Why do analysis? . . . . . . . . . . . . . . . . . . . . . . 2
2 Starting at the beginning: the natural numbers 13
2.1 The Peano axioms . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Set theory 33
3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Russell’s paradox (Optional) . . . . . . . . . . . . . . . . 46
3.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Images and inverse images . . . . . . . . . . . . . . . . . 56
3.5 Cartesian products . . . . . . . . . . . . . . . . . . . . . 62
3.6 Cardinality of sets . . . . . . . . . . . . . . . . . . . . . . 67
4 Integers and rationals 74
4.1 The integers . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 The rationals . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Absolute value and exponentiation . . . . . . . . . . . . . 86
4.4 Gaps in the rational numbers . . . . . . . . . . . . . . . . 90
5 The real numbers 94
5.1 Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Equivalent Cauchy sequences . . . . . . . . . . . . . . . . 100
5.3 The construction of the real numbers . . . . . . . . . . . 102
5.4 Ordering the reals . . . . . . . . . . . . . . . . . . . . . . 111
vii
viii Contents
5.5 The least upper bound property . . . . . . . . . . . . . . 116
5.6 Real exponentiation, part I . . . . . . . . . . . . . . . . . 121
6 Limits of sequences 126
6.1 Convergence and limit laws . . . . . . . . . . . . . . . . . 126
6.2 The Extended real number system . . . . . . . . . . . . . 133
6.3 Suprema and Infima of sequences . . . . . . . . . . . . . 137
6.4 Limsup, Liminf, and limit points . . . . . . . . . . . . . . 139
6.5 Some standard limits . . . . . . . . . . . . . . . . . . . . 148
6.6 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.7 Real exponentiation, part II . . . . . . . . . . . . . . . . 152
7 Series 155
7.1 Finite series . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.2 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.3 Sums of non-negative numbers . . . . . . . . . . . . . . . 170
7.4 Rearrangement of series . . . . . . . . . . . . . . . . . . . 174
7.5 The root and ratio tests . . . . . . . . . . . . . . . . . . . 178
8 Infinite sets 181
8.1 Countability . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2 Summation on infinite sets . . . . . . . . . . . . . . . . . 188
8.3 Uncountable sets. . . . . . . . . . . . . . . . . . . . . . . 195
8.4 The axiom of choice . . . . . . . . . . . . . . . . . . . . . 198
8.5 Ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . 202
9 Continuous functions on R 211
9.1 Subsets of the real line . . . . . . . . . . . . . . . . . . . 211
9.2 The algebra of real-valued functions . . . . . . . . . . . . 217
9.3 Limiting values of functions . . . . . . . . . . . . . . . . 220
9.4 Continuous functions . . . . . . . . . . . . . . . . . . . . 227
9.5 Left and right limits . . . . . . . . . . . . . . . . . . . . . 231
9.6 The maximum principle . . . . . . . . . . . . . . . . . . . 234
9.7 The intermediate value theorem . . . . . . . . . . . . . . 238
9.8 Monotonic functions . . . . . . . . . . . . . . . . . . . . . 241
9.9 Uniform continuity . . . . . . . . . . . . . . . . . . . . . 243
9.10 Limits at infinity. . . . . . . . . . . . . . . . . . . . . . . 249
10 Differentiation of functions 251
10.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . 251
Contents ix
10.2 Local maxima, local minima, and derivatives . . . . . . . 257
10.3 Monotone functions and derivatives . . . . . . . . . . . . 260
10.4 Inverse functions and derivatives . . . . . . . . . . . . . . 261
10.5 L’Hˆopital’s rule . . . . . . . . . . . . . . . . . . . . . . . 264
11 The Riemann integral 267
11.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
11.2 Piecewise constant functions . . . . . . . . . . . . . . . . 272
11.3 Upper and lower Riemann integrals . . . . . . . . . . . . 276
11.4 Basic properties of the Riemann integral . . . . . . . . . 280
11.5 Riemann integrability of continuous functions . . . . . . 285
11.6 Riemann integrability of monotone functions . . . . . . . 289
11.7 A non-Riemann integrable function . . . . . . . . . . . . 291
11.8 The Riemann-Stieltjes integral . . . . . . . . . . . . . . . 292
11.9 The two fundamental theorems of calculus . . . . . . . . 295
11.10 Consequences of the fundamental theorems . . . . . . . . 300
A Appendix: the basics of mathematical logic 305
A.1 Mathematical statements . . . . . . . . . . . . . . . . . . 306
A.2 Implication . . . . . . . . . . . . . . . . . . . . . . . . . . 312
A.3 The structure of proofs . . . . . . . . . . . . . . . . . . . 317
A.4 Variables and quantifiers . . . . . . . . . . . . . . . . . . 320
A.5 Nested quantifiers . . . . . . . . . . . . . . . . . . . . . . 324
A.6 Some examples of proofs and quantifiers . . . . . . . . . 327
A.7 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
B Appendix: the decimal system 331
B.1 The decimal representation of natural numbers . . . . . . 332
B.2 The decimal representation of real numbers . . . . . . . . 335
Index 339
Texts andReadingsinMathematics 349
Preface to the second and third editions
Since the publication of the first edition, many students and lectur-
ers have communicated a number of minor typos and other corrections
to me. There was also some demand for a hardcover edition of the
texts. Because of this, the publishers and I have decided to incorporate
the corrections and issue a hardcover second edition of the textbooks.
The layout, page numbering, and indexing of the texts have also been
changed; in particular the two volumes are now numbered and indexed
separately. However, the chapter and exercise numbering, as well as the
mathematical content, remains the same as the first edition, and so the
two editions can be used more or less interchangeably for homework and
study purposes.
Thethirdeditioncontainsanumberofcorrectionsthatwerereported
forthesecondedition,togetherwithafewnewexercises,butisotherwise
essentially the same text.
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