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

✍ Scribed by Kit-Wing Yu

• Year: 2021
• Tongue: English
• Leaves: 620
• Category: Library

✦ Synopsis

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:

1. 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.
2. There are 40 illustrations for explaining the mathematical concepts or ideas used behind the questions or theorems.
3. Sections in each chapter are added so as to increase the readability of the exercises.
4. 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)
5. Necessary lemmas with proofs are provided because some questions require additional mathematical concepts which are not covered by Rudin.
6. Many useful or relevant references are provided to some questions for your future research.

✦ Table of Contents

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]