Real Analysis with Real Applications

For one/two-semester undergraduate courses in Real Analysis. Using a progressive but flexible format, this text develops the principles of real analysis and shows how they can be used in a wide variety of applications. The theory and principles of real analysis are treated thoroughly, including basic and advanced examples. The applications form more than half of the book. Each is developed in its own chapter, and they show the wide range of classical and modern subjects in which real analysis is used. Building on students' knowledge of calculus and linear algebra, this text prepares them both for more intensive study of each application and for more advanced work in analysis.

Table of Contents:
1. Background. The Language of Mathematics. Sets and Functions. Calculus. Linear Algebra. The Role of Proofs. Appendix: Equivalence Relations. A. ABSTRACT ANALYSIS. 2. The Real Numbers. An Overview of the Real Numbers. Infinite Decimals. Limits. Basic Properties of Limits. Upper and Lower Bounds. Subsequences. Cauchy Sequences. Appendix: Cardinality. 3. Series. Convergent Series. Convergence Tests for Series. The Number e. Absolute and Conditional Convergence. 4. The Topology of Rn. n-dimensional Space. Convergence and Completeness in Rn. Closed and Open Subsets of Rn. Compact Sets and the Heine-Borel Theorem. 5. Functions. Limits and Continuity. Discontinuous Functions. Properties of Continuous Functions. Compactness and Extreme Values. Uniform Continuity. The Intermediate Value Theorem. Monotone Functions. 6. Normed Vector Spaces. Differentiable Functions. The Mean Value Theorem. Riemann Integration. The Fundamental Theorem of Calculus. Wallis's Product and Stirling's Formula. Measure Zero and Lebesgue's Theorem. 7. Differentiation and Integration. Definition and Examples. Topology in Normed Spaces. Inner Product Spaces. Orthonormal Sets. Orthogonal Expansions in Inner Product Spaces. Finite-Dimensional Normed Spaces. The LP Norms. 8. Limits of Functions. Limits of Functions. Uniform Convergence and Continuity. Uniform Convergence and Integration. Series of Functions. Power Series. Compactness and Subsets of C(K). 9. Metric Spaces. Definitions and Examples. Compact Metric Spaces. Complete Metric Spaces. Connectedness. Metric Completion. The LP Spaces and Abstract Integration. B. APPLICATIONS. 10. Approximation by Polynomials. Taylor Series. How Not to Approximate a Function. Bernstein's Proof of the Weierstrass Theorem. Accuracy of Approximation. Existence of Best Approximations. Characterizing Best Approximations. Expansions Using Chebychev Polynomials. Splines. Uniform Approximation by Splines. Appendix: The Stone-Weierstrass Theorem. 11. Discrete Dynamical Systems. Fixed Points and the Contraction Principle. Newton's Method. Orbits of a Dynamical System. Periodic Points. Chaotic Systems. Topological Conjugacy. Iterated Function Systems and Fractals. 12. Differential Equations. Integral Equations and Contractions. Calculus of Vector-Valued Functions. Differential Equations and Fixed Points. Solutions of Differential Equations. Local Solutions. Linear Differential Equations. Perturbation and Stability of Des. Existence without Uniqueness. 13. Fourier Series and Physics. The Steady-State Heat Equation. Formal Solution. Orthogonality Relations. Convergence in the Open Disk. The Poisson Formula. Poisson's Theorem. The Maximum Principle. The Vibrating String (Formal Solution). The Vibrating String (Rigourous Solution). Appendix: The Complex Exponential. 14. Fourier Series and Approximation. Least Squares Approximations. The Isoperimetric Problem. The Riemann-Lebesgue Lemma. Pointwise Convergence of Fourier Series. Gibbs's Phenomenon. Cesaro Summation of Fourier Series. Best Approximation by Trig Polynomials. Connections with Polynomial Approximation. Jackson's Theorem and Bernstein's Theorem. 15. Wavelets. Introduction. The Haar Wavelet. Multiresolution Analysis. Recovering the Wavelet. Daubechies Wavelets. Existence of the Daubechies Wavelets. Approximations Using Wavelets. The Franklin Wavelet. Riesz Multiresolution Analysis. 16. Convexity and Optimization. Convex Sets. Relative Interior. Separation Theorems. Extreme Points. Convex Functions in One Dimension. Convex Functions in Higher Dimensions. Subdifferentials and Directional Derivatives. Tangent and Normal Cones. Constrained Minimization. The Minimax Theorem. References. Index.