A Course in Real Analysis

by McDonald; Weiss

• ISBN13: 9780127428307
• ISBN10: 0127428305
• Format: Hardcover
• Copyright: 1999-01-12
• Publisher: Elsevier Science
• More Book Details
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Summary

A Course in Real Analysis provides a firm foundation in real analysis concepts and principles while presenting a broad range of topics in a clear and concise manner. This student-oriented text balances theory and applications, and contains a wealth of examples and exercises. Throughout the text, the authors adhere to the idea that most students learn more efficiently by progressing from the concrete to the abstract. McDonald and Weiss have also created real application chapters on probability theory, harmonic analysis, and dynamical systems theory. The text offers considerable flexibility in the choice of material to cover. * Motivation of Key Concepts: The importance of and rationale behind key ideas are made transparent * Illustrative Examples: Roughly 200 examples are presented to illustrate definitions and results * Abundant and Varied Exercises: Over 1200 exercises are provided to promote understanding * Biographies: Each chapter begins with a brief biography of a famous mathematician

Table of Contents

Preface	xiii
PART ONE Set Theory, Real Numbers, and Calculus
Set Theory	2	(32)
Georg Cantor
Basic Definitions and Properties	3	(9)
Functions and Sets	12	(8)
Equivalence of Sets; Countability	20	(6)
Algebras, σ-Algebras, and Monotone Classes	26	(8)
The Real Number System and Calculus	34	(58)
Georg Friedrich Bernhard Riemann
The Real Number System	35	(8)
Sequences of Real Numbers	43	(14)
Open and Closed Sets	57	(8)
Real-Valued Functions	65	(8)
The Cantor Set and Cantor Function	73	(8)
The Riemann Integral	81	(11)
PART TWO Measure, Integration, and Differentiation
Lebesgue Theory on the Real Line	92	(74)
Emile Felix-Edouard-Justin Borel
Borel Measurable Functions and Borel Sets	93	(10)
Lebesgue Outer Measure	103	(7)
Further Properties of Lebesgue Outer Measure	110	(8)
Lebesgue Measure	118	(10)
The Lebesgue Integral for Nonnegative Functions	128	(12)
Convergence Properties of the Lebesgue Integral for Nonnegative Functions	140	(9)
The General Lebesgue Integral	149	(12)
Lebesgue Almost Everywhere	161	(5)
Measure Theory	166	(94)
Henri Leon Lebesgue
Measure Spaces	167	(7)
Measurable Functions	174	(10)
The Abstract Lebesgue Integral for Nonnegative Functions	184	(8)
The General Abstract Lebesgue Integral	192	(11)
Convergence in Measure	203	(4)
Extensions to Measures	207	(13)
The Lebesgue-Stieltjes Integral	220	(11)
Product Measure Spaces	231	(14)
Iteration of Integrals in Product Measure Spaces	245	(15)
Elements of Probability	260	(54)
Andrei Nikolaevich Kolmogorov
The Mathematical Model for Probability	262	(12)
Random Variables	274	(14)
Expectation of Random Variables	288	(13)
The Law of Large Numbers	301	(13)
Differentiation	314	(96)
Johann Radon
Derivatives and Dini-Derivates	315	(15)
Functions of Bounded Variation	330	(4)
The Indefinite Lebesgue Integral	334	(8)
Absolutely Continuous Functions	342	(12)
Signed Measures	354	(10)
The Radon-Nikodym Theorem	364	(13)
Signed and Complex Measures	377	(13)
Decomposition of Measures	390	(12)
Measurable Transformations and the General Change-of-Variable Formula	402	(8)
PART THREE Topological, Metric, and Normed Spaces
Elements of Topological, Metric, and Normed Spaces	410	(82)
Pavel Samuilovich Urysohn
Introduction to Topological Spaces	411	(8)
Metrics and Norms	419	(8)
Weak Topologies	427	(4)
Closed Sets, Convergence, and Completeness	431	(7)
Nets and Continuity	438	(9)
Separation Properties	447	(6)
Connected Sets	453	(6)
Separability, Second Countability, and Metrizability	459	(5)
Compact Metric Spaces	464	(7)
Compact Topological Spaces	471	(4)
Locally Compact Spaces	475	(6)
Function Spaces	481	(11)
Complete Spaces, Compact Spaces, and Approximation	492	(34)
Marshall Harvey Stone
The Baire Category Theorem	493	(5)
Contractions of Complete Metric Spaces	498	(5)
Compactness in the Space C(Ω,Λ)	503	(6)
Compactness of Product Spaces	509	(4)
Approximation by Functions From a Lattice	513	(5)
Approximation by Functions From an Algebra	518	(8)
Hilbert Spaces and the Classical Banach Spaces	526	(52)
David Hilbert
Preliminaries on Normed Spaces	527	(6)
Hilbert Spaces	533	(12)
Bases and Duality in Hilbert Spaces	545	(8)
Lp-Spaces	553	(10)
Nonnegative Linear Functionals on C(Ω)	563	(10)
The Dual Spaces of C(Ω) and C0(Ω)	573	(5)
Basic Theory of Normed and Locally Convex Spaces	578	(56)
Stefan Banach
The Hahn-Banach Theorem	579	(11)
Linear Opeators on Banach Spaces	590	(7)
Topological Linear Spaces	597	(12)
Weak and Weak* Topologies	609	(9)
Compact Convex Sets	618	(16)
PART FOUR Harmonic Analysis and Dynamical Systems
Elements of Harmonic Analysis	634	(62)
Ingrid Daubechies
Introduction to Fourier Series	636	(7)
Convergence of Fourier Series	643	(10)
The Fourier Transform	653	(9)
Fourier Transforms of Measures	662	(10)
L2-Theory of the Fourier Transform	672	(6)
Introduction to Wavelets	678	(6)
Orthonormal Wavelet Bases; The Wavelet Transform	684	(12)
Measurable Dynamical Systems	696	(37)
Claude Elwood Shannon
Introduction and Examples	697	(10)
Ergodic Theory	707	(8)
Isomorphism of Measurable Dynamical Systems; Entropy	715	(8)
The Kolmogorov-Sinai Theorem; Calculation of Entropy	723	(10)
Index	733