Real and Functional AnalysisThis book is meant as a text for a first year graduate course in analysis. Any standard course in undergraduate analysis will constitute sufficient preparation for its understanding, for instance, my Undergraduate Anal ysis. I assume that the reader is acquainted with notions of uniform con vergence and the like. In this third edition, I have reorganized the book by covering inte gration before functional analysis. Such a rearrangement fits the way courses are taught in all the places I know of. I have added a number of examples and exercises, as well as some material about integration on the real line (e.g. on Dirac sequence approximation and on Fourier analysis), and some material on functional analysis (e.g. the theory of the Gelfand transform in Chapter XVI). These upgrade previous exercises to sections in the text. In a sense, the subject matter covers the same topics as elementary calculus, viz. linear algebra, differentiation and integration. This time, however, these subjects are treated in a manner suitable for the training of professionals, i.e. people who will use the tools in further investiga tions, be it in mathematics, or physics, or what have you. In the first part, we begin with point set topology, essential for all analysis, and we cover the most important results. |
Contents
| 3 | |
| 17 | |
3 Compact Spaces | 31 |
CHAPTER III | 51 |
CHAPTER XVIII | 57 |
PART TWO Banach and Hilbert Spaces | 63 |
CHAPTER IV | 64 |
2 Banach Algebras | 72 |
5 Exercises | 326 |
PART FOUR Calculus | 329 |
Differential Calculus | 331 |
2 The Derivative as a Linear Map | 333 |
3 Properties of the Derivative | 335 |
4 Mean Value Theorem | 340 |
5 The Second Derivative | 343 |
6 Higher Derivatives and Taylors Formula | 346 |
3 The Linear Extension Theorem | 75 |
4 Completion of a Normed Vector Space | 76 |
5 Spaces with Operators | 81 |
Convex Sets | 83 |
Mazurs Theorem | 88 |
6 Exercises | 91 |
CHAPTER V | 95 |
2 Functionals and Operators | 104 |
3 Exercises | 107 |
PART THREE Integration | 109 |
The General Integral | 111 |
1 Measured Spaces Measurable Maps and Positive Measures | 112 |
2 The Integral of Step Maps | 126 |
3 The LıCompletion | 128 |
First Part | 134 |
Second Part | 137 |
6 Approximations | 147 |
7 Extension of Positive Measures from Algebras to σAlgebras | 153 |
8 Product Measures and Integration on a Product Space | 158 |
9 The Lebesgue Integral in Rş | 166 |
10 Exercises | 172 |
CHAPTER VII | 181 |
2 Duality Between Lıµ and Lμ | 185 |
3 Complex and Vectorial Measures | 195 |
4 Complex or Vectorial Measures and Duality | 204 |
5 The Lş Spaces 1 p 8 | 209 |
6 The Law of Large Numbers | 213 |
7 Exercises | 217 |
Some Applications of Integration | 223 |
2 Continuity and Differentiation Under the Integral Sign | 225 |
3 Dirac Sequences | 227 |
4 The Schwartz Space and Fourier Transform | 236 |
5 The Fourier Inversion Formula | 241 |
6 The Poisson Summation Formula | 243 |
7 An Example of Fourier Transform Not in the Schwartz Space | 244 |
8 Exercises | 247 |
CHAPTER IX | 251 |
1 Positive and Bounded Functionals on CX | 252 |
2 Positive Functionals as Integrals | 255 |
3 Regular Positive Measures | 265 |
4 Bounded Functionals as Integrals | 267 |
5 Localization of a Measure and of the Integral | 269 |
CHAPTER XXIII | 270 |
6 Product Measures on Locally Compact Spaces | 272 |
7 Exercises | 274 |
CHAPTER X | 278 |
2 Applications to Fourier Analysis | 287 |
3 Exercises | 294 |
CHAPTER XI | 295 |
2 Support and Localization | 299 |
3 Derivation of Distributions | 303 |
4 Distributions with Discrete Support | 304 |
CHAPTER XII | 308 |
2 The Haar Integral Uniqueness | 313 |
3 Existence of the Haar Integral | 319 |
4 Measures on Factor Groups and Homogeneous Spaces | 322 |
7 Partial Derivatives | 351 |
8 Differentiating Under the Integral Sign | 355 |
9 Differentiation of Sequences | 356 |
10 Exercises | 357 |
CHAPTER XIV | 360 |
2 The Implicit Mapping Theorem | 364 |
3 Existence Theorem for Differential Equations | 365 |
4 Local Dependence on Initial Conditions | 371 |
5 Global Smoothness of the Flow | 376 |
6 Exercises | 379 |
PART FIVE Functional Analysis | 385 |
CHAPTER XV | 387 |
2 Orthogonality | 391 |
3 Applications of the Open Mapping Theorem | 395 |
CHAPTER XVI | 400 |
2 The Gelfand Transform | 407 |
3 CAlgebras | 409 |
4 Exercises | 412 |
CHAPTER XVII | 415 |
2 Fredholm Operators and the Index | 417 |
3 Spectral Theorem for Compact Operators | 426 |
4 Application to Integral Equations | 432 |
5 Exercises | 433 |
Spectral Theorem for Bounded Hermitian Operators | 438 |
2 Positive Hermitian Operators | 439 |
3 The Spectral Theorem for Compact Hermitian Operators | 442 |
4 The Spectral Theorem for Hermitian Operators | 444 |
5 Orthogonal Projections | 449 |
6 Schurs Lemma | 453 |
8 The MorsePalais Lemma | 457 |
9 Exercises | 458 |
CHAPTER XIX | 464 |
2 SelfAdjoint Operators | 469 |
The Laplace Operator in the Plane | 476 |
CHAPTER XX | 480 |
the TitchmarshKodaira Formula | 485 |
3 Unbounded Functions of Operators | 488 |
4 Spectral Families of Projections | 490 |
5 The Spectral Integral as Stieltjes Integral | 491 |
6 Exercises | 492 |
PART SIX Global Analysis | 495 |
CHAPTER XXI | 497 |
2 Change of Variables Formula | 498 |
3 Differential Forms | 507 |
4 Inverse Image of a Form | 512 |
5 Appendix | 516 |
CHAPTER XXII | 523 |
2 Submanifolds | 527 |
3 Tangent Spaces | 533 |
4 Partitions of Unity | 536 |
5 Manifolds with Boundary | 539 |
6 Vector Fields and Global Differential Equations | 543 |
Integration and Measures on Manifolds | 547 |
6 Stokes Theorem with Singularities | 561 |
| 574 | |
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Common terms and phrases
A₁ assume Banach space bilinear bounded Cauchy sequence Chapter closed sets closed subspace closure compact set complete complex numbers concludes the proof contained continuous functions continuous linear map continuous maps converges almost everywhere converges pointwise convex set Corollary countable define definition denote denumerable disjoint dominated convergence theorem du(x Exercise exists f₁ finite measure finite number follows function f given Haar measure Hence Hilbert space inequality integral intersection interval inverse isomorphism lemma Let f Let g locally compact map f measurable sets measured space metric space non-empty normed vector space o-algebra o-finite open sets partition polynomial positive measure properties proves our theorem real numbers respect satisfies seminorm sets of finite Show step functions step maps sup norm topological space topology u-measurable unique v₁ weak topology whence x₁