Real Analysis Vol. 1 : Modern Techniques and Their Applications

by Folland, Gerald B.

• ISBN13: 9780471317166
• ISBN10: 0471317160
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-04-07
• Publisher: Wiley

Summary

* Updated material on Hausdorff dimension and fractal dimension.

Author Biography

GERALD B. FOLLAND is Professor of Mathematics at the University of Washington in Seattle. He has written extensively on mathematical analysis, including Fourier analysis, harmonic analysis, and differential equations.

Table of Contents

Preface	vii
Prologue	1	(18)
The Language of Set Theory	1	(3)
Orderings	4	(2)
Cardinality	6	(3)
More about Well Ordered Sets	9	(1)
The Extended Real Number System	10	(3)
Metric Spaces	13	(3)
Notes and References	16	(3)
Measures	19	(24)
Introduction	19	(2)
σ-algebras	21	(3)
Measures	24	(4)
Outer Measures	28	(5)
Borel Measures on the Real Line	33	(7)
Notes and References	40	(3)
Integration	43	(42)
Measurable Functions	43	(6)
Integration of Nonnegative Functions	49	(3)
Integration of Complex Functions	52	(8)
Modes of Convergence	60	(4)
Product Measures	64	(6)
The n-dimensional Lebesgue Integral	70	(7)
Integration in Polar Coordinates	77	(4)
Notes and References	81	(4)
Signed Measures and Differentiation	85	(28)
Signed Measures	85	(3)
The Lebesgue-Radon-Nikodym Theorem	88	(5)
Complex Measures	93	(2)
Differentiation on Euclidean Space	95	(5)
Functions of Bounded Variation	100	(9)
Notes and References	109	(4)
Point Set Topology	113	(38)
Topological Spaces	113	(6)
Continuous Maps	119	(6)
Nets	125	(3)
Compact Spaces	128	(3)
Locally Compact Hausdorff Spaces	131	(5)
Two Compactness Theorems	136	(2)
The Stone-Weierstrass Theorem	138	(5)
Embeddings in Cubes	143	(3)
Notes and References	146	(5)
Elements of Functional Analysis	151	(30)
Normed Vector Spaces	151	(6)
Linear Functionals	157	(4)
The Baire Category Theorem and its Consequences	161	(4)
Topological Vector Spaces	165	(6)
Hilbert Spaces	171	(8)
Notes and References	179	(2)
Lp Spaces	181	(30)
Basic Theory of Lp Spaces	181	(7)
The Dual of Lp	188	(5)
Some Useful Inequalities	193	(4)
Distribution Functions and Weak Lp	197	(3)
Interpolation of Lp Spaces	200	(8)
Notes and References	208	(3)
Radon Measures	211	(24)
Positive Linear Functionals on Cc(X)	211	(5)
Regularity and Approximation Theorems	216	(5)
The Dual of Co(X)	221	(5)
Products of Radon Measures	226	(5)
Notes and References	231	(4)
Elements of Fourier Analysis	235	(46)
Preliminaries	235	(4)
Convolutions	239	(8)
The Fourier Transform	247	(10)
Summation of Fourier Integrals and Series	257	(6)
Pointwise Convergence of Fourier Series	263	(7)
Fourier Analysis of Measures	270	(3)
Applications to Partial Differential Equations	273	(5)
Notes and References	278	(3)
Elements of Distribution Theory	281	(32)
Distributions	281	(10)
Compactly Supported, Tempered, and Periodic Distributions	291	(10)
Sobolev Spaces	301	(9)
Notes and References	310	(3)
Topics in Probability Theory	313	(26)
Basic Concepts	313	(7)
The Law of Large Numbers	320	(5)
The Central Limit Theorem	325	(3)
Construction of Sample Spaces	328	(2)
The Wiener Process	330	(6)
Notes and References	336	(3)
More Measures and Integrals	339	(26)
Topological Groups and Haar Measure	339	(9)
Hausdorff Measures	348	(7)
Self-similarity and Hausdorff Dimension	355	(6)
Integration on Manifolds	361	(2)
Notes and References	363	(2)
Bibliography	365	(12)
Index of Notation	377	(2)
Index	379