Analysis 2

235 Pages·2016·1.26 MB·English

by  Tao, Terence

Texts and Readings in Mathematics 38
Terence Tao
Analysis II
Third Edition
Texts and Readings in Mathematics
Volume 38
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 II
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 ix
Preface to the first edition xi
About the Author xvii
1 Metric spaces 1
1.1 Definitions and examples . . . . . . . . . . . . . . . . . . 1
1.2 Some point-set topology of metric spaces . . . . . . . . . 10
1.3 Relative topology . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Cauchy sequences and complete metric spaces . . . . . . 17
1.5 Compact metric spaces . . . . . . . . . . . . . . . . . . . 21
2 Continuous functions on metric spaces 28
2.1 Continuous functions . . . . . . . . . . . . . . . . . . . . 28
2.2 Continuity and product spaces . . . . . . . . . . . . . . . 31
2.3 Continuity and compactness . . . . . . . . . . . . . . . . 34
2.4 Continuity and connectedness . . . . . . . . . . . . . . . 36
2.5 Topological spaces (Optional) . . . . . . . . . . . . . . . 39
3 Uniform convergence 45
3.1 Limiting values of functions . . . . . . . . . . . . . . . . 46
3.2 Pointwise and uniform convergence . . . . . . . . . . . . 48
3.3 Uniform convergence and continuity . . . . . . . . . . . . 53
3.4 The metric of uniform convergence . . . . . . . . . . . . 56
3.5 Series of functions; the Weierstrass M-test . . . . . . . . 58
3.6 Uniform convergence and integration . . . . . . . . . . . 61
3.7 Uniform convergence and derivatives . . . . . . . . . . . 63
3.8 Uniform approximation by polynomials . . . . . . . . . . 66
4 Power series 75
4.1 Formal power series . . . . . . . . . . . . . . . . . . . . . 75
4.2 Real analytic functions . . . . . . . . . . . . . . . . . . . 78
vii
viii Contents
4.3 Abel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Multiplication of power series . . . . . . . . . . . . . . . 87
4.5 The exponential and logarithm functions . . . . . . . . . 90
4.6 A digression on complex numbers . . . . . . . . . . . . . 93
4.7 Trigonometric functions . . . . . . . . . . . . . . . . . . . 101
5 Fourier series 107
5.1 Periodic functions . . . . . . . . . . . . . . . . . . . . . . 108
5.2 Inner products on periodic functions . . . . . . . . . . . . 110
5.3 Trigonometric polynomials . . . . . . . . . . . . . . . . . 114
5.4 Periodic convolutions . . . . . . . . . . . . . . . . . . . . 116
5.5 The Fourier and Plancherel theorems . . . . . . . . . . . 121
6 Several variable differential calculus 127
6.1 Linear transformations . . . . . . . . . . . . . . . . . . . 127
6.2 Derivatives in several variable calculus . . . . . . . . . . 134
6.3 Partial and directional derivatives . . . . . . . . . . . . . 137
6.4 The several variable calculus chain rule . . . . . . . . . . 144
6.5 Double derivatives and Clairaut’s theorem . . . . . . . . 147
6.6 The contraction mapping theorem . . . . . . . . . . . . . 149
6.7 The inverse function theorem in several variable calculus 152
6.8 The implicit function theorem . . . . . . . . . . . . . . . 157
7 Lebesgue measure 162
7.1 The goal: Lebesgue measure . . . . . . . . . . . . . . . . 164
7.2 First attempt: Outer measure . . . . . . . . . . . . . . . 166
7.3 Outer measure is not additive . . . . . . . . . . . . . . . 174
7.4 Measurable sets . . . . . . . . . . . . . . . . . . . . . . . 176
7.5 Measurable functions . . . . . . . . . . . . . . . . . . . . 183
8 Lebesgue integration 187
8.1 Simple functions . . . . . . . . . . . . . . . . . . . . . . . 187
8.2 Integration of non-negative measurable functions . . . . . 193
8.3 Integration of absolutely integrable functions . . . . . . . 201
8.4 Comparison with the Riemann integral . . . . . . . . . . 205
8.5 Fubini’s theorem . . . . . . . . . . . . . . . . . . . . . . . 207
Index 213
Texts andReadingsinMathematics 219
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.
ix
Preface to the first edition
This text originated from the lecture notes I gave teaching the honours
undergraduate-level real analysis sequence at the University of Califor-
nia, Los Angeles, in 2003. Among the undergraduates here, real anal-
ysis was viewed as being one of the most difficult courses to learn, not
only because of the abstract concepts being introduced for the first time
(e.g., topology, limits, measurability, etc.), but also because of the level
of rigour and proof demanded of the course. Because of this percep-
tion of difficulty, one was often faced with the difficult choice of either
reducing the level of rigour in the course in order to make it easier, or
to maintain strict standards and face the prospect of many undergradu-
ates, even many of the bright and enthusiastic ones, struggling with the
course material.
Faced with this dilemma, I tried a somewhat unusual approach to
thesubject. Typically,anintroductorysequenceinrealanalysisassumes
thatthestudentsarealreadyfamiliarwiththerealnumbers, withmath-
ematical induction, with elementary calculus, and with the basics of set
theory, and then quickly launches into the heart of the subject, for in-
stance the concept of a limit. Normally, students entering this sequence
do indeed have a fair bit of exposure to these prerequisite topics, though
in most cases the material is not covered in a thorough manner. For in-
stance, very few students were able to actually define a real number, or
even an integer, properly, even though they could visualize these num-
bers intuitively and manipulate them algebraically. This seemed to me
to be a missed opportunity. Real analysis is one of the first subjects
(together with linear algebra and abstract algebra) that a student en-
counters, in which one truly has to grapple with the subtleties of a truly
rigorous mathematical proof. As such, the course offered an excellent
chance to go back to the foundations of mathematics, and in particular
xi