About the Book:
The third edition of this well known text continues to provide a solid foundation inmathematical analysis for undergraduate and first-year graduate students. The textbegins with a discussion of the real number system as a complete ordered field.(Dedekind's construction is now treated in an appendix to Chapter I.) The topologicalbackground needed for the development of convergence, continuity, differentiation andintegration is provided in Chapter 2. There is a new section on the gamma function, andmany new and interesting exercises are included.
About the Author:
Walter Rudin, UNIV OF WISC - MADISON
Chapter 1: The Real and Complex Number SystemsIntroduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix ExercisesChapter 2: Basic TopologyFinite, Countable, and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets ExercisesChapter 3: Numerical Sequences and SeriesConvergent 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 ExercisesChapter 4: ContinuityLimits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity ExercisesChapter 5: DifferentiationThe 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 ExercisesChapter 6: The Riemann-Stieltjes IntegralDefinition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves ExercisesChapter 7: Sequences and Series of FunctionsDiscussion 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 ExercisesChapter 8: Some Special FunctionsPower Series The Exponential and Logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier Series The Gamma Function ExercisesChapter 9: Functions of Several VariablesLinear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem Determinants Derivatives of Higher Order Differentiation of Integrals ExercisesChapter 10: Integration of Differential FormsIntegration Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains Stokes' Theorem Closed Forms and Exact Forms Vector Analysis ExercisesChapter 11: The Lebesgue TheorySet 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 L2 ExercisesBibliography List of Special Symbols Index