A Complete Solution Guide to Real and Complex Analysis (Walter Rudin's)

This is a complete solution guide to all exercises from Chapters 1 to 20 in Rudin's Real and Complex Analysis . The features of this book are as follows: It covers all the 397 exercises from Chapters 1 to 20 with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. There are 40 illustrations 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 are provided because some questions require additional mathematical concepts which are not covered by Rudin. Many useful or relevant references are provided to some questions for your future research. A Complete Solution Guide to Walter Rudin's "Real and Complex Analysis" I Title page About the author Preface List of Figures Contents Chapter 1. Abstract Integration 1.1 Problems on σ-algebras and Measurable Functions 1.2 Problems related to the Lebesgue’s MCT/DCT Chapter 2. Positive Borel Measure 2.1 Properties of Semicontinuity 2.2 Problems on the Lebesgue Measure on R 2.3 Integration of Sequences of Continuous Functions 2.4 Problems on Borel Measures and Lebesgue Measures 2.5 Problems on Regularity of Borel Measures 2.6 Miscellaneous Problems on L^1 and Other Properties Chapter 3. L^p-Spaces 3.1 Properties of Convex Functions 3.2 Relations among Lp-Spaces and some Consequences 3.3 Applications of Theorems 3.3, 3.5, 3.8, 3.9 and 3.12 3.4 Hardy’s Inequality and Egoroff’s Theorem 3.5 Convergence in Measure and the Essential Range of f ∈ L^∞(μ) 3.6 A Converse of Jensen’s Inequality 3.7 The Completeness/Completion of a Metric Space 3.8 Miscellaneous Problems Chapter 4. Elementary Hilbert Space Theory 4.1 Basic Properties of Hilbert Spaces 4.2 Application of Theorem 4.14 4.3 Miscellaneous Problems Chapter 5. Examples on Banach Space Techniques 5.1 The Unit Ball in a Normed Linear Space 5.2 Failure of Theorem 4.10 and Norm-preserving Extensions 5.3 The Dual Space of X 5.4 Applications of Baire’s and other Theorems 5.5 Miscellaneous Problems Chapter 6. Complex Measures 6.1 Properties of Complex Measures 6.2 Dual Spaces of L^p(μ) 6.3 Fourier Coefficients of Complex Borel Measures 6.4 Problems on Uniformly Integrable Sets 6.5 Dual Spaces of L^p(μ) Revisit Chapter 7. Differentiation 7.1 Lebesgue Points and Metric Densities 7.2 Periods of Functions and Lebesgue Measurable Groups 7.3 The Cantor Function and the Non-measurability of f ◦ T 7.4 Problems related to the AC of a Function 7.5 Miscellaneous Problems on Differentiation Chapter 8. Integration on Product Spaces 8.1 Monotone Classes and Ordinate Sets of Functions 8.2 Applications of the Fubini Theorem 8.3 The Product Measure Theorem and Sections of a Function 8.4 Miscellaneous Problems Chapter 9. Fourier Transforms 9.1 Properties of The Fourier Transforms 9.2 The Poisson Summation Formula and its Applications 9.3 Fourier Transforms on R^k and its Applications Index Bibliography [1]-[18] [19]-[38] [39]-[60] [61]-[68] A Complete Solution Guide to Walter Rudin's "Real and Complex Analysis" II Preface List of Figures Chapter 10. Elementary Properties of Holomorphic Functions 10.1 Basic Properties of Holomorphic Functions 10.2 Evaluation of Integrals 10.3 Composition of Holomorphic Functions and Morera's Theorem 10.4 Problems related to Zeros of Holomorphic Functions 10.5 Laurent Series and its Applications 10.6 Miscellaneous Problems Chapter 11. Harmonic Functions 11.1 Basic Properties of Harmonic Functions 11.2 Harnack's Inequalities and Positive Harmonic Functions 11.3 The Weak* Convergence and Radial Limits of Holomorphic Functions 11.4 Miscellaneous Problems Chapter 12. The Maximum Modulus Principle 12.1 Applications of the Maximum Modulus Principle 12.2 Asymptotic Values of Entire Functions 12.3 Further Applications of the Maximum Modulus Principle Chapter 13. Approximations by Rational Functions 13.1 Meromorphic Functions on S2 and Applications of Runge's Theorem 13.2 Holomorphic Functions in the Unit Disc without Radial Limits 13.3 Simply Connectedness and Miscellaneous Problems Chapter 14. Conformal Mapping 14.1 Basic Properties of Conformal Mappings 14.2 Problems on Normal Families and the Class S 14.3 Proofs of Conformal Equivalence between Annuli 14.4 Constructive Proof of the Riemann Mapping Theorem Chapter 15. Zeros of Holomorphic Functions 15.1 Infinite Products and the Order of Growth of an Entire Function 15.2 Some Examples 15.3 Problems on Blaschke Products 15.4 Miscellaneous Problems and the Müntz-Szasz Theorem Chapter 16. Analytic Continuation 16.1 Singular Points and Continuation along Curves 16.2 Problems on the Modular Group and Removable Sets 16.3 Miscellaneous Problems Chapter 17. H^p-Spaces 17.1 Problems on Subharmonicity and Harmonic Majoriants 17.2 Basic Properties of H^p 17.3 Factorization of f ∈ H^p 17.4 A Projection of L^p onto H^p 17.5 Miscellaneous Problems Chapter 18. Elementary Theory of Banach Algebras 18.1 Examples of Banach Spaces and Spectrums 18.2 Properties of Ideals and Homomorphisms 18.3 The Commutative Banach algebra H^∞ Chapter 19. Holomorphic Fourier Transforms 19.1 Problems on Entire Functions of Exponential Type 19.2 Quasi-analytic Classes and Borel's Theorem Chapter 20. Uniform Approximation by Polynomials Index Bibliography [1]-[17] [18]-[38] [39]-[59] [60]-[84]