A complete solution guide to principles of mathematical analysis

This is a complete solution guide to all exercises in Rudin's Principles of Mathematical Analysis . The features of this book are as It covers all the 285 exercises with detailed and completed solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. There are 55 illustrations and 3 tables for explaining the mathematical concepts or ideas used behind the questions or theorems. Sections in each chapter are added so as to increase the readability of the exercises. Different colors are used frequently in order to highlight or explain problems, lemmas, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) Necessary lemmas with proofs and references are provided because some questions require additional mathematical concepts which are not covered by Rudin. Three appendices are included which further explain and supplement some theories in Chapters 10 and 11. Cover 1 Rudin(PMA) 2 Preface 7 List of Figures 9 List of Tables 11 The Real and Complex Number Systems 16 Problems on rational numbers and fields 16 Properties of supremums and infimums 17 An index law and the logarithm 17 Properties of the complex field 20 Properties of Euclidean spaces 22 A supplement to the proof of Theorem 1.19 24 Basic Topology 26 The empty set and properties of algebraic numbers 26 The uncountability of irrational numbers 26 Limit points, open sets and closed sets 27 Some metrics 30 Compact sets 32 Further topological properties of R 33 Properties of connected sets 36 Separable metric spaces and bases and a special case of Baire's theorem 39 Numerical Sequences and Series 46 Problems on sequences 46 Problems on series 48 Recursion formulas of sequences 59 A representation of the Cantor set 64 Cauchy sequences and the completions of metric spaces 65 Continuity 72 Properties of continuous functions 72 The extension, the graph and the restriction of a continuous function 73 Problems on uniformly continuous functions 77 Further properties of continuous functions 83 Discontinuous functions 84 The distance function E 88 Convex functions 91 Other properties of continuous functions 96 Differentiation 100 Problems on differentiability of a function 100 Applications of Taylor's theorem 111 Derivatives of higher order and iteration methods 117 Solutions of differential equations 128 The Riemann-Stieltjes Integral 132 Problems on Riemann-Stieltjes integrals 132 Definitions of improper integrals 137 Hölder's inequality 140 Problems related to improper integrals 145 Applications and a generalization of integration by parts 148 Problems on rectifiable curves 152 Sequences and Series of Functions 156 Problems on uniform convergence of sequences of functions 156 Problems on equicontinuous families of functions 172 Applications of the (Stone-)Weierstrass theorem 179 Isometric mappings and initial-value problems 182 Some Special Functions 188 Problems related to special functions 188 Index of a curve 216 Stirling's formula 224 Functions of Several Variables 228 Linear transformations 228 Differentiable mappings 230 Local maxima and minima 234 The inverse function theorem and the implicit function theorem 240 The rank of a linear transformation 252 Derivatives of higher order 256 Integration of Differential Forms 270 Integration over sets in Rk and primitive mappings 270 Generalizations of partitions of unity 278 Applications of Theorem 10.9 (Change of Variables Theorem) 282 Properties of k-forms and k-simplexes 299 Problems on closed forms and exact forms 309 Problems on vector fields 344 The Lebesgue Theory 352 Further properties of integrable functions 352 The Riemann integrals and the Lebesgue integrals 355 Functions of classes L and L2 360 Appendix 369 A proof of Lemma 10.14 370 Solid angle subtended by a surface at the origin 380 Proofs of some basic properties of a measure 384 Index 392 Bibliography 394