Counterexamples on Uniform Convergence - Andrei Bourchtein

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Biographie:
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Sommaire:
Preface ix List of Examples xi List of Figures xxix About the Companion Website xxxiii Introduction xxxv I.1 Comments xxxv I.1.1 On the Structure of This Book xxxv I.1.2 On Mathematical Language and Notation xxxvii I.2 Background (Elements of Theory) xxxviii I.2.1 Sequences of Functions xxxviii I.2.2 Series of Functions xli I.2.3 Families of Functions xliv 1 Conditions of Uniform Convergence 1 1.1 Pointwise, Absolute, and Uniform Convergence. Convergence on a Set and Subset 1 1.2 Uniform Convergence of Sequences and Series of Squares and Products 15 1.3 Dirichlet's and Abel's Theorems 31 Exercises 39 Further Reading 42 2 Properties of the Limit Function: Boundedness, Limits, Continuity 45 2.1 Convergence and Boundedness 45 2.2 Limits and Continuity of Limit Functions 51 2.3 Conditions of Uniform Convergence. Dini's Theorem 68 2.4 Convergence and Uniform Continuity 79 Exercises 88 Further Reading 93 3 Properties of the Limit Function: Differentiability and Integrability 95 3.1 Differentiability of the Limit Function 95 3.2 Integrability of the Limit Function 117 Exercises 128 Further Reading 131 4 Integrals Depending on a Parameter 133 4.1 Existence of the Limit and Continuity 133 4.2 Differentiability 144 4.3 Integrability 154 Exercises 162 Further Reading 166 5 Improper Integrals Depending on a Parameter 167 5.1 Pointwise, Absolute, and Uniform Convergence 167 5.2 Convergence of the Sum and Product 176 5.3 Dirichlet's and Abel's Theorems 185 5.4 Existence of the Limit and Continuity 192 5.5 Differentiability 198 5.6 Integrability 202 Exercises 210 Further Reading 214 Bibliography 215 Index 217