## Advanced Real AnalysisAdvanced 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

1 | |

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 |

451 | |

Index of Notation | 455 |

459 | |