Chapter 1 The Real and Complex Number Systems
1.1 Introduction 1
1.2 The field axioms . 1
1.3 The order axioms 2
1.4 Geometric representation of real numbers 3
1.5 Intervals 3
1.6 Integers 4
1.7 The unique factorization theorem for integers 4
1.8 Rational numbers 6
1.9 Irrational numbers 7
1.10 Upper bounds, maximum element, least upper bound(supremum) . 8
1.11 The comleess axiom 9
1.12 Some properties of the supremum 9
1.13 Properties of the integers deduce fo the comleess axim 0
1.14 The Archimedean property of the real-number system . 10
1.15 Rational numbers with finite decimal representation 11
1.16 Finite decimal approximations to real numbers 11
1.17 Infinite decimal representation of real numbers . 12
1.18 Absolute values and the triangle inequality 12
1.19 The Cauchy—Schwarz inequality 13
1.20 Plus and minus infinity and the extended real number system R* 14
1.21 Complex numbers 15
1.22 Geometric representation of complex numbers 17
1. The imaginary unit 18
1.24 Absolute value of a complex number . 18
1.25 Isibility of ordering the complex numbers . 19
1.26 Complex exponentials 19
1.27 Further properties of complex exponentials 20
1.28 The argument of a complex number . 20
1.29 Integral powers and roots of complex numbers . 21
1.30 Complex logarithms 22
1.31 Complex powers
1.32 Complex sines and cosines 24
1.33 Infinity and the extended complex plane C* 24
Exercises 25
Chapter 2 Some Basic Notions of Set Theory
2.1 Introductiou 32
2.2 Notations 32
. Ordered pairs 33
2.4 Cartesian product of two sets 33
2.5 Relations and functions 34
2.6 Further terminology concerning functions 35
2.7 One-to-one functions and inverses 36
2.8 Coite functions 37
2.9 Sequences. 38
2.10 Similar (equinumerous) sets 38
2.11 Finite and infinite sets 39
2.12 Countable and uncountable sets 39
2.13 Uncountability of the real-number system 42
2.14 Set algebra 43
2.15 Countable collections of countable sets
Exercises 43
Chapter 3 Elements of Point Set Topology
3.1 Introduction 47
3.2 Euclidean space Rt 47
3.3 Open balls and open sets in R* 49
3.4 The structure of open sets in RH 50
3.5 Closed sets . 52
3.6 Adhèrent points. Accumulation points 52
3.7 Closed sets and adhèrent points 53
3.8 The Bolzano—Weierstrass theorem 54
3.9 The Cantor intersection theorem 56
3.10 The Lindel?f covering theorem 56
3.11 The Heine—Borel covering theorem 58
3.12 Compactness in R‘ 59
3.13 Metric spaces 60
3.14 Point set topology in metric spaces 61
3.15 Compact subsets of a metric space 63
3.16 Boundary of a set
Exercises 65
Chaqter 4 LimisndCntinuity
4.1 Introduction 70
4.2 Convergent sequences in a metric space 72
4.3 Cauchy sequences 74
4.4 Complete metric spaces . 74
4.5 Limit of a function 76
4.6 Limits of complex-valued functions
4.7 Limits of vector-valued functions 77
4.8 Continuous functions 78
4.9 Continuity of coite functions.
4.10 Continuous complex-valued and vector-valued functions 79
4.11 Examples of continuous functions 80
4.12 Continuity and inverse images of open or closed sets 80
4.13 Functions continuous on compact sets 81
4.14 Topolo$ical mappings (homeomorphisms) 82
4.15 Bolzano’s theorem 84
4.16 Connectedness 84
4.17 Components of a metric space . 86
4.18 Arcwise connectedness 87
4.19 Uniform continuity 88
4.20 Uniform continuiyndcmpact sets 90
4.21 Fixed-point theorem for contractions 91
4.22 Discontinuities of real-valued functions 92
4. Monotonic functions 94
Exercises 95
Chapter 5 DerJvatives
5.1Introduction 104
5.2 Definition of derivative .104
5.3 Derivatives and continuity 105
5.4 Algebra of derivatives106
5.5 The chain rule 106
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汤姆·M. 阿波斯托尔(Tom M. Apostol)是加州理工学院数学系荣誉教授。他于1946年在华盛顿大学西雅图分校获得数学硕士，于1948年在加州大学伯克利分校获得数学博士。
本书是一部现代数学名著。自20世纪70年代面世以来，一直受到西方学术界、教育界的广泛推崇，被许多知名大学指定为教材。
本书是在“高等微积分”的水平上阐述数学分析中的论题，提供了从初等微积分向实变函数论及复变函数论中的高等课程的一种过渡，而且介绍了某些涉及现代分析的抽象理论．内容既涵盖我国大学的数学分析课程的内容，又包括勒贝格积分及柯西定理和留数计算等.
本书条理清晰，内容精练，言简意赅，适合作为高等院校生数学分析课程的教材．
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