Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics

Description

Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and $n$-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series. The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed. Cover Title page Copyright Contents Preface Chapter 0. Where We’re Starting and Where We’re Going Chapter 1. Essential Tools 1.1. Sets and statements 1.2. Functions 1.3. Countability and uncountability 1.4. Induction 1.5. Order in the real line 1.6. Some vital inequalities 1.7. Exercises Chapter 2. Metric Spaces 2.1. The definition of a metric space 2.2. Important metrics in Rⁿ 2.3. Open balls and open sets in metric spaces 2.4. Closed sets and limit points 2.5. Interior, closure, and boundary 2.6. Dense subsets 2.7. Equivalent metrics 2.8. Normed vector spaces 2.9. A brief note about conventions 2.10. Exercises Chapter 3. Sequences 3.1. Convergence 3.2. Discrete dynamical systems 3.3. Sequences and limit points 3.4. Algebraic theorems for sequences 3.5. Subsequences 3.6. Completeness 3.7. The contraction mapping principle 3.8. Sets of sequences as metric spaces 3.9. Exercises Chapter 4. Continuity 4.1. The definition of continuity 4.2. Equivalent formulations of continuity 4.3. Continuity and limit theorems for scalar- valued functions 4.4. Continuity and products of metric spaces 4.5. Uniform continuity 4.6. The metric space 𝐶([𝑎,𝑏],R ) 4.7. An application to functional equations 4.8. Exercises Chapter 5. Compactness and Connectedness 5.1. Basic definitions and results on compactness 5.2. The nested set property for compact sets 5.3. Compactness and continuity 5.4. Other facts about compactness 5.5. Connectedness 5.6. Periodic points of maps on intervals 5.7. Injective continuous functions defined on intervals 5.8. Exercises Chapter 6. The Derivative 6.1. The definition of the derivative 6.2. Differentiation rules 6.3. Applications of the derivative 6.4. Exercises Chapter 7. The Riemann Integral 7.1. Partitions and the definition of the integral 7.2. Basic properties of the integral 7.3. The fundamental theorem of calculus 7.4. Ordinary differential equations 7.5. Exercises Chapter 8. Sequences of Functions 8.1. Infinite series 8.2. Power series 8.3. Higher derivatives and Taylor polynomials 8.4. Differentiation and integration of sequences of functions 8.5. The exponential function 8.6. Compact subsets in 𝐶[𝑎,𝑏] 8.7. Exercises Chapter 9. Chaos in Discrete Dynamical Systems 9.1. The definition of chaos 9.2. Semiconjugacy 9.3. Subshifts of finite type 9.4. Itineraries and piecewise expanding maps 9.5. A dynamical system with a dense orbit but no periodic points 9.6. Exercises Chapter 10. The Hausdorff Metric and Fractals 10.1. Definition of the Hausdorff metric 10.2. Properties of the Hausdorff metric 10.3. Fractals in the plane 10.4. Exercises Bibliography Index Back Cover