Introduction to Analysis

Cover......Page 1
Title: Introduction to Analysis......Page 2
ISBN 0-488-65038-3......Page 3
Preface......Page 4
Contents......Page 6
Introduction to Analysis......Page 8
CHAPTER 1:Notions from Set Theory......Page 10
§ 1. SETS AND ELEMENTS. SUBSETS......Page 11
§ 2. OPERATIONS ON SETS.......Page 13
§ 3. FUNCI\'IONS......Page 17
§ 4. FINITE AND INFINITE SETS.......Page 19
PROBLEMS......Page 21
CHAPTER II:The Real Number System......Page 23
§ 1. THE FIELD PROPERTIES.......Page 24
§ 2. ORDER......Page 27
§ 3. THE LEAST UPPER BOUND PROPERTY......Page 31
§ 4.THE EXISTENCE OF SQUARE ROOTS.......Page 36
PROBUNS......Page 37
CHAPTER III: Metric Spaces......Page 40
§ 1. DEFINITION OF METRIC SPACE. EXAMPLES.......Page 41
§ 2. OPEN AND CLOSED SETS.......Page 44
§ 3. CONVERGENT SEQUENCES......Page 51
§ 4. COMPLETENESS......Page 58
§ 5. COMPACTNESS......Page 61
§ 6. CONNECTEDNESS......Page 66
PROBLEMS......Page 68
CHAPTER IV: Continuous Functions......Page 73
§ 1. DEFINITION OF CONTINUITY. EXAMPLES......Page 74
§ 2. CONTINUITY AND LIMITS......Page 78
§ 3. THE CONTINUITY OF RATIONAL OPERATIONS. FUNCTIONS WITH VALUES IN E^n......Page 81
§ 4. CONTINUOUS FUNCTIONS ON A COMPACT METRIC SPACE......Page 84
§ 5. CONTINUOUS FUNCTIONS ON A CONNECTED METRIC SPACE......Page 88
§ 6. SEQUENCES OF FUNCTIONS......Page 89
PROBLEMS......Page 96
CHAPTER V: Differentiation......Page 102
§ 1. THE DEFINITION OF DERIVATIVE.......Page 103
§ 2. RULES OF DIFFERENTIATION......Page 105
§ 3.THE MEAN VALUE THEOREM.......Page 108
§ 4. TAYLOR\'S THEOREM......Page 111
PROBLEMS......Page 113
CHAPTER VI: Riemann Integration......Page 116
§ 1. DEFINITIONS AND EXAMPLES......Page 117
§ 2  LINEARITY AND ORDER PROPERTIES OF THE INTEGRAL......Page 121
§ 3. EXISTENCE OF THE INTEGRAL......Page 123
§ 4. THE FUNDAMENTAL THEOREM OF CALCULUS......Page 128
§ 5. THE LOGARITHMIC AND EXPONENTIAL FUNCTONS......Page 133
PROBLEMS......Page 137
CHAPTER VII: Interchage of Limit Operations......Page 141
§ 1. INTEGRATION AND DIFFERENTIATION OF SEQUENCES OF FUNCTIONS......Page 142
§ 2. INFINITE SERIES........Page 145
§ 3. POWER SERIES......Page 154
§ 4. THE TRIGONOMETRIC FUNCI\'lONS......Page 160
§ 5. DIFFERENTIATION UNDER THE INTEGRAL SIGN......Page 163
PROBLEMS......Page 164
CHAPTER VIII: The Method of Successive Approximations......Page 172
§ 1. THE FIXED POINT THEOREM......Page 173
§ 2. THE SIMPLEST CASE OF THE IMPLICIT FUNCTION THEOREM......Page 176
§ 3. EXISTENCE AND UNIQUENESS.THEOREMS FOR ORDINARY DIFFERINTIAL EQUATIONS.......Page 180
PROBLEMS......Page 193
CHAPTER IX: Partial Differentiation......Page 196
§ 1. DEFINITIONS AND BASIC PROPERTIES.......Page 197
§ 2. HIGHER DERIVATIVES......Page 204
§ 3. THE IMPLICIT FUNCTION THEOREM......Page 208
PROBLEMS......Page 215
CHAPTER X: Multiple Integrals......Page 218
§ 1. RIEMANN INTEGRATION ON A CLOSED INTERVAL IN E^n. EXAMPLES AND BASIC PROPERTIES......Page 219
§ 2. EXISTENCE OF THE INTEGRAL. INTEGRATION ON ARBITRARY SUBSETS OF E^n. VOLUME......Page 225
§ 3. ITERATED INTEGRALS......Page 0
§ 4. CHANGE OF VARIABLE.......Page 238
PROBLEMS......Page 247
Suggestions for Further Reading......Page 252
Index......Page 253