Mathematical analysis is often referred to as generalized calculus. But it is much more than that. This book has been written in the belief that emphasizing the inherent nature of a mathematical discipline helps students to understand it better. With this in mind, and focusing on the essence of analysis, the text is divided into two parts based on the way they are related to calculus: completion and abstraction. The first part describes those aspects of analysis which complete a corresponding area of calculus theoretically, while the second part concentrates on the way analysis generalizes some aspects of calculus to a more general framework. Presenting the contents in this way has an important advantage: students first learn the most important aspects of analysis on the classical space and fill in the gaps of their calculus-based knowledge. Then they proceed to a step-by-step development of an abstract theory, namely, the theory of metric spaces which studies such crucial notions as limit, continuity, and convergence in a wider context.
The readers are assumed to have passed courses in one- and several-variable calculus and an elementary course on the foundations of mathematics. A large variety of exercises and the inclusion of informal interpretations of many results and examples will greatly facilitate the reader's study of the subject.
Hossein Hosseini Giv, University of Sistan and Baluchestan, Zahedan, Iran
To the Instructor To the Student Introduction and Outline of the Book Acknowledgments
Part 1. Rebuilding the Calculus Building Chapter 1. The Real Number System Revisited 1.1. The Algebraic Axioms 1.2. The Order Axioms 1.3. Absolute Value, Distance, and Neighborhoods 1.4. Natural Numbers and Mathematical Induction 1.5. The Axiom of Completeness and Its Uses 1.6. The Complex Number System Notes on Essence and Generalizability Exercises
Chapter 2. Sequences and Series of Real Numbers 2.1. Real Sequences, Their Convergence, and Boundedness 2.2. Subsequences, Limit Superior and Limit Inferior 2.3. Cauchy Sequences 2.4. Sequences in Closed and Bounded Intervals 2.5. Series: Revisiting Some Convergence Tests 2.6. Rearrangements of Series 2.7. Power Series Notes on Essence and Generalizability Exercises
Chapter 3. Limit and Continuity of Real Functions 3.1. Limit Points and Some Other Classes of Points in ā„¯ 3.2. A More General Definition of Limit 3.3. Limit at Infinity 3.4. One-Sided Limits\index{one-sided!limit} 3.5. Continuity and Two Kinds of Discontinuity 3.6. Continuity on [š¯‘ˇ, š¯‘¸]: Results and Applications 3.7. Uniform Continuity Notes on Essence and Generalizability Exercises
Chapter 4. Derivative and Differentiation 4.1. The Why and What of the Concept of Derivative 4.2. The Basic Properties of Derivative 4.3. Local Extrema and Derivative 4.4. The Mean Value Theorem: More Applications of Derivative 4.5. Taylor Series: A First Glance 4.6. Taylorā€™s Theorem and the Convergence of Taylor Series Notes on Essence and Generalizability Exercises
Chapter 5. The Riemann Integral 5.1. Motivation: The Area Problem 5.2. The Riemann Integral: Definition and Basic Results 5.3. Some Integrability Theorems 5.4. Antiderivatives and the Fundamental Theorem of Calculus Notes on Essence and Generalizability Exercises
Part 2. Abstraction and Generalization Chapter 6. Basic Theory of Metric Spaces 6.1. A First Generalization: The Definition of Metric Space 6.2. Neighborhoods and Some Classes of Points 6.3. Open and Closed Sets 6.4. Metric Subspaces 6.5. Boundedness and Total Boundedness Notes on Essence and Generalizability Exercises
Chapter 7. Sequences in General Metric Spaces 7.1. Convergence and Divergence in Metric Spaces 7.2. Cauchy Sequences and Complete Metric Spaces 7.3. Compactness: Definition and Some Basic Results 7.4. Compactness: Some Equivalent Forms 7.5. Perfect Sets and Cantorā€™s Set Notes on Essence and Generalizability Exercises
Chapter 8. Limit and Continuity of Functions in Metric Spaces 8.1. The Definition of Limit in General Metric Spaces 8.2. Continuity and Uniform Continuity 8.3. Continuity and Compactness 8.4. Connectedness and Its Relation to Continuity 8.5. Banachā€™s Fixed Point Theorem Notes on Essence and Generalizability Exercises
Chapter 9. Sequences and Series of Functions 9.1. Sequences of Functions and Their Pointwise Convergence 9.2. Uniform Convergence 9.3. Weierstrassā€™s Approximation Theorem 9.4. Series of Functions and Their Convergence Notes on Essence and Generalizability Exercises Appendix Real Sequences and Series Limit and Continuity of Functions The Concepts of Derivative and Differentiability The Riemann Integral Bibliography Index