Description
Book SynopsisA comprehensive and thorough analysis of concepts and results on uniform convergence
Counterexamples on Uniform Convergence: Sequences, Series, Functions, and Integrals presents counterexamples to false statements typically found within the study of mathematical analysis and calculus, all of which are related to uniform convergence. The book includes the convergence of sequences, series and families of functions, and proper and improper integrals depending on a parameter. The exposition is restricted to the main definitions and theorems in order to explore different versions (wrong and correct) of the fundamental concepts and results.
The goal of the book is threefold. First, the authors provide a brief survey and discussion of principal results of the theory of uniform convergence in real analysis. Second, the book aims to help readers master the presented concepts and theorems, which are traditionally challenging and are sources of misunderstanding and
Trade Review
"The features of the book include An overview of important concepts and theorems on uniform convergence, Well-organized coverage of the majority of the topics on uniform convergence studied in analysis courses, An original approach to the analysis of important results on uniform convergence
based on counterexamples, Additional exercises at varying levels of complexity for each topic covered in the book & A supplementary Instructor's Solutions Manual containing complete solutions to
all exercises, which is available via a companion website" Mathematical Reviews, Sept 2017
Table of Contents
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
