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 completeness axiom 9
1.12 Some properties of the supremum 9
1.13 Properties of the integers deduced from the completeness axiom 10
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
1.23 The imaginary unit 18
1.24 Absolute value of a complex number . 18
1.25 Impossibility 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 23
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
2.3 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 Composite 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 R't 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 Limits and Continuity
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 composite 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 continuity and compact sets 90
4.21 Fixed-point theorem for contractions 91
4.22 Discontinuities of real-valued functions 92
4.23 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
5.6 One-si