Companion to Real Analysis

Companion to Real Analysis John M. Erdman

Portland State University Version November 20, 2012

c 2007 John M. Erdman This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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E-mail address: erdman@pdx.edu

Contents

PREFACE

vii

Greek Letters

ix

Fraktur Fonts

x

Chapter 1. SETS

1

1.1. Set Notation

1

1.2. Families of Sets

1

1.3. Subsets

2

1.4. Unions and Intersections

2

1.5. Complements

3

1.6. Symmetric Difference

4

1.7. Notation for Sets of Numbers

5

Chapter 2. FUNCTIONS

7

2.1. Cartesian Products

7

2.2. Relations

7

2.3. Functions

8

2.4. Images and Inverse Images

9

2.5. Composition of Functions

9

2.6. The Identity Function

9

2.7. Diagrams

10

2.8. Some Special Functions

10

2.9. Injections, Surjections, and Bijections

11

2.10. Inverse Functions

12

2.11. Equivalence Relations and Quotients

14

Chapter 3. CARDINALITY

17

3.1. Finite and Infinite Sets

17

3.2. Countable and Uncountable Sets

18

Chapter 4. GROUPS, VECTOR SPACES, AND ALGEBRAS

21

4.1. Operations

21

4.2. Groups

22

4.3. Homomorphisms of Semigroups and Groups

23

4.4. Vector Spaces

25

4.5. Linear Transformations

26

4.6. Rings and Algebras

28

4.7. Ring and Algebra Homomorphisms

30

Chapter 5. PARTIALLY ORDERED SETS

33

5.1. Partial and Linear Orderings

33

5.2. Infima and Suprema

34

5.3. Zorn's Lemma

35

5.4. Lattices

37

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CONTENTS

5.5. Lattice Homomorphisms

39

5.6. Boolean Algebras

39

Chapter 6. THE REAL NUMBERS

43

6.1. Axioms Defining the Real Numbers

43

6.2. Construction of the Real Numbers

44

6.3. Elementary Functions

47

6.4. Absolute Value

47

6.5. Some Useful Inequalities

48

6.6. Complex Numbers

49

Chapter 7. SEQUENCES AND INDEXED FAMILIES

51

7.1. Sequences

51

7.2. Indexed Families of Sets

52

7.3. Limit Inferior and Limit Superior (for Sets)

53

7.4. Limit Inferior and Limit Superior (for Real Numbers)

54

7.5. Subsequences and Cluster Points

58

Chapter 8. CATEGORIES

59

8.1. Objects and Morphisms

59

8.2. Quotients

61

8.3. Products

63

8.4. Coproducts

65

Chapter 9. ORDERED VECTOR SPACES

67

9.1. Partially Orderings on Vector Spaces

67

9.2. Convexity

68

9.3. Positive Cones

68

9.4. Finitely Additive Set Functions

69

Chapter 10. TOPOLOGICAL SPACES

71

10.1. Definition of Topology

71

10.2. Base for a Topology

73

10.3. Some Elementary Topological Properties

74

10.4. Metric Spaces

74

10.5. Interiors and Closures

79

Chapter 11. CONTINUITY AND WEAK TOPOLOGIES

81

11.1. Continuity--the Global Property

81

11.2. Continuity--the Local Property

82

11.3. Uniform Continuity

83

11.4. Weak Topologies

84

11.5. Subspaces

85

11.6. Quotient Topologies

86

Chapter 12. NORMED LINEAR SPACES

89

12.1. Norms

89

12.2. Bounded Linear Maps

92

12.3. Products of Normed Linear Spaces

94

12.4. Quotients of Normed Linear Spaces

96

Chapter 13. DIFFERENTIATION

97

13.1. Tangency

97

13.2. The Differential

98

CONTENTS

v

Chapter 14. RIESZ SPACES

101

14.1. Definition and Elementary Properties

101

14.2. Riesz Homomorphisms and Positive Operators

104

14.3. The Order Dual of a Riesz Space

106

Chapter 15. MEASURABLE SPACES

109

15.1. -Algebras of Sets

109

15.2. Borel Sets

110

15.3. Measurable Functions

111

15.4. Functors

112

Chapter 16. THE RIESZ SPACE OF REAL MEASURES

115

16.1. Real Measures

115

16.2. Ideals in Riesz Spaces

117

16.3. Bands in Riesz spaces

119

16.4. Nets

119

16.5. Disjointness in Riesz Spaces

123

16.6. Absolute Continuity

125

Chapter 17. COMPACT SPACES

127

17.1. Compactness

127

17.2. Local Compactness

130

17.3. Compactifications

131

Chapter 18. LEBESGUE MEASURE

135

18.1. Positive Measures

135

18.2. Outer Measures

136

18.3. Lebesgue Measure on R

138

18.4. The Space L(S)

140

Chapter 19. THE LEBESGUE INTEGRAL

143

19.1. Integration of Simple Functions

143

19.2. Integration of Positive Functions

145

19.3. Integration of Real and Complex Valued Functions

147

Chapter 20. COMPLETE METRIC SPACES

151

20.1. Cauchy Sequences

151

20.2. Completions and Universal Morphisms

153

20.3. Compact Subsets of C(X)

155

20.4. Banach Spaces; Lp-spaces

156

20.5. Banach Algebras

158

20.6. Hilbert spaces

159

Chapter 21. ALGEBRAS AND LATTICES OF CONTINUOUS FUNCTIONS

163

21.1. Banach Lattices

163

21.2. The Stone-Weierstrass Theorems

164

21.3. Semicontinuous Functions

165

21.4. Normal Topological Spaces

166

21.5. The Hahn-Tong-Katetov Theorem

167

21.6. Ideals in C(X)

169

Chapter 22. FUNCTIONS OF BOUNDED VARIATION

171

22.1. Preliminaries on Monotone Functions

171

22.2. Variation

172

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