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Summary: Explains set theory, sequences, continuity, differentiation, integrals, and vector-space concepts.Edition/Copyright: 3RD 76
Rudin, Walter : University of Wisconsin
Chapter 1: The Real and Complex Number Systems
The Real Field
The Extended Real Number System
The Complex Field
Chapter 2: Basic Topology
Finite, Countable, and Uncountable Sets
Chapter 3: Numerical Sequences and Series
Upper and Lower Limits
Some Special Sequences
Series of Nonnegative Terms
The Number e
The Root and Ratio Tests
Summation by Parts
Addition and Multiplication of Series
Chapter 4: Continuity
Limits of Functions
Continuity and Compactness
Continuity and Connectedness
Infinite Limits and Limits at Infinity
Chapter 5: Differentiation
The Derivative of a Real Function
Mean Value Theorems
The Continuity of Derivatives
Derivatives of Higher-Order
Differentiation of Vector-valued Functions
Chapter 6: The Riemann-Stieltjes Integral
Definition and Existence of the Integral
Properties of the Integral
Integration and Differentiation
Integration of Vector-valued Functions
Chapter 7: Sequences and Series of Functions
Discussion of Main Problem
Uniform Convergence and Continuity
Uniform Convergence and Integration
Uniform Convergence and Differentiation
Equicontinuous Families of Functions
The Stone-Weierstrass Theorem
Chapter 8: Some Special Functions
The Exponential and Logarithmic Functions
The Trigonometric Functions
The Algebraic Completeness of the Complex Field
The Gamma Function
Chapter 9: Functions of Several Variables
The Contraction Principle
The Inverse Function Theorem
The Implicit Function Theorem
The Rank Theorem
Derivatives of Higher Order
Differentiation of Integrals
Chapter 10: Integration of Differential Forms
Partitions of Unity
Change of Variables
Simplexes and Chains
Closed Forms and Exact Forms
Chapter 11: The Lebesgue Theory
Construction of the Lebesgue Measure
Comparison with the Riemann Integral
Integration of Complex Functions
Functions of Class L²
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