9780471317166

Real Analysis Vol. 1 : Modern Techniques and Their Applications

by
  • ISBN13:

    9780471317166

  • ISBN10:

    0471317160

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 4/7/1999
  • Publisher: Wiley

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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

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