Introduction to Real Analysis: An Educational Approach
July 2009, ©2009
Bridging the gap between the development and history of real analysis, Introduction to Real Analysis: An Educational Approach presents a comprehensive introduction to real analysis while also offering a survey of the field. With its balance of historical background, key calculus methods, and hands-on applications, this book provides readers with a solid foundation and fundamental understanding of real analysis.
The book begins with an outline of basic calculus, including a close examination of problems illustrating links and potential difficulties. Next, a fluid introduction to real analysis is presented, guiding readers through the basic topology of real numbers, limits, integration, and a series of functions in natural progression. The book moves on to analysis with more rigorous investigations, and the topology of the line is presented along with a discussion of limits and continuity that includes unusual examples in order to direct readers' thinking beyond intuitive reasoning and on to more complex understanding. The dichotomy of pointwise and uniform convergence is then addressed and is followed by differentiation and integration. Riemann-Stieltjes integrals and the Lebesgue measure are also introduced to broaden the presented perspective. The book concludes with a collection of advanced topics that are connected to elementary calculus, such as modeling with logistic functions, numerical quadrature, Fourier series, and special functions.
Detailed appendices outline key definitions and theorems in elementary calculus and also present additional proofs, projects, and sets in real analysis. Each chapter references historical sources on real analysis while also providing proof-oriented exercises and examples that facilitate the development of computational skills. In addition, an extensive bibliography provides additional resources on the topic.
Introduction to Real Analysis: An Educational Approach is an ideal book for upper- undergraduate and graduate-level real analysis courses in the areas of mathematics and education. It is also a valuable reference for educators in the field of applied mathematics.
1 Elementary Calculus.
1.1 Preliminary Concepts.
1.2 Limits and Continuity.
1.5 Sequences and Series of Constants.
1.6 Power Series and Taylor Series.
Interlude: Fermat, Descartes, and theTangent Problem.
2 Introduction to Real Analysis.
2.1 Basic Topology of the Real Numbers.
2.2 Limits and Continuity.
2.4 Riemann and Riemann-Stieltjes Integration.
2.5 Sequences, Series, and Convergence Tests.
2.6 Pointwise and Uniform Convergence.
Interlude: Euler and the "Basel Problem".
3 A Brief Introduction to Lebesgue Theory.
3.1 Lebesgue Measure and Measurable Sets.
3.2 The Lebesgue Integral.
3.3 Measure, Integral, and Convergence.
3.4 Littlewood’s Three Principles.
Interlude: The Set of Rational Numbers isVery Large andVery Small.
4 Special Topics.
4.1 Modeling with Logistic Functions—Numerical Derivatives.
4.2 Numerical Quadrature.
4.3 Fourier Series.
4.4 Special Functions—The Gamma Function.
4.5 Calculus Without Limits: Differential Algebra.
Appendix A: Definitions and Theorems of Elementary Real Analysis.
A.3 The Derivative.
A.4 Riemann Integration.
A.5 Riemann-Stieltjes Integration.
A.6 Sequences and Series of Constants.
A.7 Sequences and Series of Functions.
Appendix B: A Very Brief Calculus Chronology.
Appendix C: Projects in Real Analysis.
C.1 Historical Writing Projects.
C.2 Induction Proofs: Summations, Inequalities, and Divisibility.
C.3 Series Rearrangements.
C.4 Newton and the Binomial Theorem.
C.5 Symmetric Sums of Logarithms.
C.6 Logical Equivalence: Completeness of the Real Numbers.
C.7 Vitali’s Nonmeasurable Set.
C.8 Sources for Real Analysis Projects.
C.9 Sources for Projects for Calculus Students.
- Provides a unique organization of the topic; a brief review of basic concepts of calculus is followed by an introduction to real analysis and a thorough discussion of measure theory
- Contains numerous exercises and examples that are both proof oriented and facilitate an understanding of computational skills
- Engages readers by beginning with an AP calculus focus, and then quickly moving to the more theoretical aspects of mathematical analysis topics
- Thoroughly classroom-tested material based on the author's own Real Analysis for Teachers course
- Is accompanied by supplemental material including a bibliography of calculus, analysis, and pedagogical references as well as appendices that contain the main theorems of real analysis and a collection of student projects to be used in the classroom