Advanced Real Analysis

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Springer Science & Business Media, Jul 11, 2008 - Mathematics - 466 pages
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Advanced Real Analysis systematically develops those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. Along with a companion volume Basic Real Analysis (available separately or together as a Set via the Related Links nearby), these works present a comprehensive treatment with a global view of the subject, emphasizing the connections between real analysis and other branches of mathematics.

Key topics and features of Advanced Real Analysis:

* Develops Fourier analysis and functional analysis with an eye toward partial differential equations

* Includes chapters on Sturm–Liouville theory, compact self-adjoint operators, Euclidean Fourier analysis, topological vector spaces and distributions, compact and locally compact groups, and aspects of partial differential equations

* Contains chapters about analysis on manifolds and foundations of probability

* Proceeds from the particular to the general, often introducing examples well before a theory that incorporates them

* Includes many examples and nearly two hundred problems, and a separate 45-page section gives hints or complete solutions for most of the problems

* Incorporates, in the text and especially in the problems, material in which real analysis is used in algebra, in topology, in complex analysis, in probability, in differential geometry, and in applied mathematics of various kinds

Advanced Real Analysis requires of the reader a first course in measure theory, including an introduction to the Fourier transform and to Hilbert and Banach spaces. Some familiarity with complex analysis is helpful for certain chapters. The book is suitable as a text in graduate courses such as Fourier and functional analysis, modern analysis, and partial differential equations. Because it focuses on what every young mathematician needs to know about real analysis, the book is ideal both as a course text and for self-study, especially for graduate students preparing for qualifying examinations. Its scope and approach will appeal to instructors and professors in nearly all areas of pure mathematics, as well as applied mathematicians working in analytic areas such as statistics, mathematical physics, and differential equations. Indeed, the clarity and breadth of Advanced Real Analysis make it a welcome addition to the personal library of every mathematician.

 

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Contents

INTRODUCTION TO BOUNDARYVALUE PROBLEMS
1
2 Separation of Variables
3
3 SturmLiouville Theory
19
4 Problems
31
COMPACT SELFADJOINT OPERATORS
34
2 Spectral Theorem for Compact SelfAdjoint Operators
36
3 HilbertSchmidt Theorem
41
4 Unitary Operators
45
COMPACT AND LOCALLY COMPACT GROUPS
212
1 Topological Groups
213
2 Existence and Uniqueness of Haar Measure
220
3 Modular Function
230
4 Invariant Measures on Quotient Spaces
234
5 Convolution and Lp Spaces
237
6 Representations of Compact Groups
240
7 PeterWeyl Theorem
251

5 Classes of Compact Operators
46
6 Problems
52
TOPICS IN EUCLIDEAN FOURIER ANALYSIS
54
2 Weak Derivatives and Sobolev Spaces
60
3 Harmonic Functions
69
4Hp Theory
80
5 CalderónZygmund Theorem
83
6 Applications of the CalderónZygmund Theorem
92
7 Multiple Fourier Series
96
8 Application to Traces of Integral Operators
97
9 Problems
99
TOPICS IN FUNCTIONAL ANALYSIS
105
1 Topological Vector Spaces
106
2 CU Distributions and Support
112
3 Weak and WeakStar Topologies Alaoglus Theorem
116
4 Stone Representation Theorem
121
5 Linear Functionals and Convex Sets
125
6 Locally Convex Spaces
128
7 Topology on CcomU
131
8 KreinMilman Theorem
140
9 FixedPoint Theorems
143
10 Gelfand Transform for Commutative C Algebras
146
11 Spectral Theorem for Bounded SelfAdjoint Operators
160
12 Problems
173
Distributions
179
2 Elementary Operations on Distributions
187
3 Convolution of Distributions
189
4 Role of Fourier Transform
202
5 Fundamental Solution of Laplacian
206
6 Problems
207
8 Fourier Analysis Using Compact Groups
256
9 Problems
264
ASPECTS OF PARTIAL DIFFERENTIAL EQUATIONS
275
2 Orientation
283
3 Local Solvability in the ConstantCoefficient Case
292
4 Maximum Principle in the Elliptic SecondOrder Case
296
5 Parametrices for Elliptic Equations with Constant Coefficients
300
6 Method of Psuedodifferential Operators
305
7 Problems
317
ANALYSIS ON MANIFOLDS
321
1 Differential Calculus on Smooth Manifolds
322
2 Vector Fields and Integral Curves
331
3 Identification Spaces
334
4 Vector Bundles
338
5 Distributions and Differential Operators on Manifolds
348
6 More about Euclidean Pseudodifferential Operators
355
7 Pseudodifferential Operators on Manifolds
361
8 Further Developments
366
Problems
370
FOUNDATIONS OF PROBABILITY
375
2 Independent Random Variables
381
3 Kolmogorov Extension Theorem
386
4 Strong Law of Large Numbers
393
5 Problems
399
Hints for Solutions of Problems
403
Selected References
451
Index of Notation
455
Index
459
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