Fundamental Ideas of Analysis

Michael C. Reed received the Bachelor’s degree in mathematics from Yale (1963) and the Ph.D. in mathematics from Stanford (1969). He has worked in many branches of mathematics including functional analysis, mathematical physics, partial differential equations, and the applications of mathematics to human physiology and medicine. He is currently Arts and Sciences Distinguished Professor of Mathematics at Duke University, where he uses Fundamental Ideas of Analysis to teach undergraduate and graduate students.

• PrefaceChapter 1 Preliminaries 1The Real Numbers 1Sets and Functions 6Cardinality 15Methods of Proof 20Chapter 2 Sequences 27Convergence 27Limit Theorems 35Two-state Markov Chains 40Cauchy Sequences 44Supremum and Infimum 52The Bolzano-Weierstrass Theorem 55The Quadratic Map 60Projects 68Chapter 3 The Riemann Integral 73Continuity 73Continuous Functions on Closed Intervals 80The Riemann Integral 87Numerical Methods 95Discontinuities 103Improper Integrals 113Projects 119Chapter 4 Differentiation 121Differentiable Functions 121The Fundamental Theorem of Calculus  129Taylor’s Theorem 134Newton’s Method 140Inverse Functions 147Functions of Two Variables 151Projects 159Chapter 5 Sequences of Functions 163Pointwise and Uniform Convergence 163Limit Theorems 169The Supremum Norm 175Integral Equations 183The Calculus of Variations 188Metric Spaces 196The Contraction Mapping Principle 203Normed Linear Spaces  210Projects 219Chapter 6 Series of Functions 223Lim sup and Lim inf 223Series of Real Constants 228The Weierstrass M-test 238Power Series 245Complex Numbers 252Infinite Products and Prime Numbers 260Projects 270Chapter 7 Differential Equations 273Local Existence 273Global Existence 283The Error Estimate for Euler’s Method 289Projects 296Chapter 8 Complex Analysis 299Analytic Functions 299Integration on Paths 305Cauchy's Theorem 312Projects 320Chapter 9 Fourier Series 323The Heat Equation 323Definitions and Examples 331Pointwise Convergence 337Mean-square Convergence 345Projects 355Chapter 10 Probability Theory 359Discrete Random Variables 359Coding Theory 368Continuous Random Variables 376The Variation Metric 386Projects 398Bibliography 403Symbol Index 406Index 409