Principles of Mathemetical Analysis

Chapter 1: The Real and Complex Number Systems
Introduction
Ordered Sets
Fields
The Real Field
The Extended Real Number System
The Complex Field
Euclidean Spaces
Appendix
Exercises
Chapter 2: Basic Topology
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
Summation by Parts
Absolute Convergence
Addition and Multiplication of Series
Rearrangements
Exercises
Chapter 4: Continuity
Limits of Functions
Continuous Functions
Continuity and Compactness
Continuity and Connectedness
Discontinuities
Monotonic Functions
Infinite Limits and Limits at Infinity
Exercises
Chapter 5: Differentiation
The Derivative of a Real Function
Mean Value Theorems
The Continuity of Derivatives
L'Hospital's Rule
Derivatives of Higher-Order
Taylor's Theorem
Differentiation of Vector-valued Functions
Exercises
Chapter 6: The Riemann-Stieltjes Integral
Definition and Existence of the Integral
Properties of the Integral
Integration and Differentiation
Integration of Vector-valued Functions
Rectifiable Curves
Exercises
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: Some Special Functions
Power Series
The Exponential and Logarithmic Functions
The Trigonometric Functions
The Algebraic Completeness of the Complex Field
Fourier Series
The Gamma Function
Exercises
Chapter 9: Functions of Several Variables
Linear Transformations
Differentiation
The Contraction Principle
The Inverse Function Theorem
The Implicit

Prof. Dr. Walter Rudin ist Professor (em.) an der University of Wisconsin-Madison, USA. 1949 machte er seinen Ph.D. an der Duke University, North Carolina. Anschliessend arbeitete er als Dozent am Massachusetts Institute of Technology, von wo er 1959 an die University of Wisconsin-Madison wechselte. Bekannt wurde Walter Rudin vor allem durch seine mittlerweile in 13 Sprachen übersetzten Analysis-Lehrbücher. In seinen Forschungen befasste er sich hauptsächlich mit harmonischer Analysis und komplexen Variablen. 1993 wurde ihm von der Amerikanischen Mathematischen Gesellschaft der "Leroy P. Steele Prize for Mathematical Exposition" verliehen.