# Book:Walter Rudin/Principles of Mathematical Analysis/Second Edition

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## Walter Rudin:

## Walter Rudin: *Principles of Mathematical Analysis (2nd Edition)*

Published $\text {1964}$, **McGraw-Hill Inc.**

### Subject Matter

### Contents

- Preface

- Chapter 1. The Real and Complex Number Systems

- Introduction
- Dedekind cuts
- Real numbers
- The extended real number system
- Complex numbers
- Euclidean spaces
- Exercises

- Chapter 2. Elements of Set Theory

- Finite, countable and uncountable sets
- Metric spaces
- Compact sets
- Perfect sets
- Connected sets
- Exercises

- Chapter 3. Numerical Sequences and Series

- Convergent sequences
- Subsequences
- Cauchy sequences
- Upper and lower limits
- Some special sequences
- Series
- Series of nonnegative terms
- The number $e$
- The root and ratio tests
- Power series
- Partial summation
- Absolute convergence
- Addition and multiplication of series
- Rearrangements
- Exercises

- Chapter 4. Continuity

- The limit of a function
- Continuous functions
- Continuity and compactness
- Continuity and connectedness
- Discontinuities
- Monotonic functions
- Infinite limits and limits at infinity

- Chapter 5. Differentiation

- The derivative of a real function
- Mean value theorems
- The continuity of derivatives
- L'Hospital's rule
- Derivatives of a higher order
- Taylor's theorem
- Differentiation of vector-valued functions
- Exercises

- Chapter 6. The Riemann-Stieltjes Integral

- Definition and existence of the integral
- The integral as a limit of sums
- Integration and differentiation
- Integration of vector-valued functions
- Functions of bounded variables
- Further theorems on integration

- Chapter 7. Sequences and Series of Functions

- Discussion of main problem
- Uniform convergence
- Uniform convergence and continuity
- Uniform convergence and integration
- Uniform convergence and differentiation
- Equicontinuous families of functions
- The Stone-Weierstrass theorem
- Exercises

- Chapter 8. Further Topics in the Theory of Series

- Power series
- The exponential and logarithmic functions
- The trigonometric functions
- The algebraic completeness of the complex field
- Fourier series
- Exercises

- Chapter 9. Functions of Several Variables

- Linear transformations
- Differentiation
- The inverse function theorem
- The implicit function theorem
- The rank theorem
- A decomposition theorem
- Determinants
- Integration
- Differential forms
- Simplexes and chains
- Stokes' theorem
- Exercises

- Chapter 10. The Lebesgue Theory

- Set functions
- Construction of the Lebesgue measure
- Measure spaces
- Measurable functions
- Simple functions
- Integration
- Comparison with the Riemann integral
- Integration of complex functions
- Functions of class $\LL^2$
- Exercises

- Bibliography

- List of Frequently Occurring Symbols

- Index

## Further Editions

## Cited by

## Source work progress

- 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.38$. Decimals