Real analysis

Preface......Page 3
Background and Preview......Page 12
The Real Numbers......Page 13
Compact Sets of Real Numbers......Page 18
Countable Sets......Page 21
Uncountable Cardinals......Page 24
Transfinite Ordinals......Page 27
Category......Page 30
Outer Measure and Outer Content......Page 33
Small Sets......Page 35
Measurable Sets of Real Numbers......Page 38
Nonmeasurable Sets......Page 42
Zorn\'s Lemma......Page 45
Borel Sets of Real Numbers......Page 47
Analytic Sets of Real Numbers......Page 49
Bounded Variation......Page 51
Newton\'s Integral......Page 54
Cauchy\'s Integral......Page 55
Riemann\'s Integral......Page 57
Volterra\'s Example......Page 60
Riemann--Stieltjes Integral......Page 62
Lebesgue\'s Integral......Page 65
The Generalized Riemann Integral......Page 67
Additional Problems for Chapter 1......Page 70
Measure Spaces......Page 74
One-Dimensional Lebesgue Measure......Page 75
Additive Set Functions......Page 80
Measures and Signed Measures......Page 86
Limit Theorems......Page 89
Jordan and Hahn Decomposition......Page 93
Complete Measures......Page 96
Outer Measures......Page 99
Method I......Page 103
Regular Outer Measures......Page 106
Nonmeasurable Sets......Page 110
More About Method I......Page 113
Completions......Page 116
Additional Problems for Chapter 2......Page 119
Metric Outer Measures......Page 123
Metric Space......Page 124
Metric Outer Measures......Page 127
Method II......Page 132
Approximations......Page 136
Construction of Lebesgue--Stieltjes Measures......Page 138
Properties of Lebesgue--Stieltjes Measures......Page 144
Lebesgue--Stieltjes Measures in IRn......Page 149
Hausdorff Measures and Hausdorff Dimension......Page 151
Methods III and IV......Page 158
Additional Remarks......Page 163
Additional Problems for Chapter 3......Page 167
Measurable Functions......Page 172
Definitions and Basic Properties......Page 173
Sequences of Measurable Functions......Page 178
Egoroff\'s Theorem......Page 183
Approximations by Simple Functions......Page 186
Approximation by Continuous Functions......Page 190
Additional Problems for Chapter 4......Page 195
Integration......Page 199
Introduction......Page 200
Integrals of Nonnegative Functions......Page 204
Fatou\'s Lemma......Page 208
Integrable Functions......Page 212
Riemann and Lebesgue......Page 216
Countable Additivity of the Integral......Page 224
Absolute Continuity......Page 227
Radon--Nikodym Theorem......Page 232
Convergence Theorems......Page 239
Relations to Other Integrals......Page 246
Integration of Complex Functions......Page 250
Additional Problems for Chapter 5......Page 254
Fubini\'s Theorem......Page 259
Product Measures......Page 260
Fubini\'s Theorem......Page 268
Tonelli\'s Theorem......Page 270
Additional Problems for Chapter 6......Page 272
The Vitali Covering Theorem......Page 275
Functions of Bounded Variation......Page 281
The Banach--Zarecki Theorem......Page 285
Determining a Function by Its Derivative......Page 288
Calculating a Function from Its Derivative......Page 290
Total Variation of a Continuous Function......Page 297
VBG* Functions......Page 303
Approximate Continuity, Lebesgue Points......Page 307
Additional Problems for Chapter 7......Page 313
Differentiation of Measures......Page 320
Differentiation of Lebesgue--Stieltjes Measures......Page 321
The Cube Basis; Ordinary Differentiation......Page 325
The Lebesgue Decomposition Theorem......Page 331
The Interval Basis; Strong Differentiation......Page 333
Net Structures......Page 340
Radon--Nikodym Derivative in a Measure Space......Page 346
Summary, Comments, and References......Page 354
Additional Problems for Chapter 8......Page 357
Definitions and Examples......Page 359
Convergence and Related Notions......Page 368
Continuity......Page 371
Homeomorphisms and Isometries......Page 375
Separable Spaces......Page 379
Complete Spaces......Page 381
Contraction Maps......Page 386
Applications of Contraction Mappings......Page 388
Compactness......Page 394
Totally Bounded Spaces......Page 398
Compact Sets in C(X)......Page 399
Application of the Arzelà--Ascoli Theorem......Page 403
The Stone--Weierstrass Theorem......Page 405
The Isoperimetric Problem......Page 408
More on Convergence......Page 411
Additional Problems for Chapter 9......Page 415
The Baire Category Theorem......Page 418
The Banach--Mazur Game......Page 424
The First Classes of Baire and Borel......Page 429
Properties of Baire-1 Functions......Page 434
Topologically Complete Spaces......Page 438
Applications to Function Spaces......Page 442
Additional Problems for Chapter 10......Page 453
Analytic Sets......Page 459
Products of Metric Spaces......Page 460
Baire Space......Page 461
Analytic Sets......Page 464
Borel Sets......Page 468
An Analytic Set That Is Not Borel......Page 472
Measurability of Analytic Sets......Page 474
The Suslin Operation......Page 476
A Method to Show a Set Is Not Borel......Page 478
Differentiable Functions......Page 481
Additional Problems for Chapter 11......Page 485
Normed Linear Spaces......Page 488
Compactness......Page 494
Linear Operators......Page 498
Banach Algebras......Page 502
The Hahn--Banach Theorem......Page 505
Improving Lebesgue Measure......Page 509
The Dual Space......Page 515
The Riesz Representation Theorem......Page 518
Separation of Convex Sets......Page 524
An Embedding Theorem......Page 529
The Uniform Boundedness Principle......Page 531
An Application to Summability......Page 534
The Open Mapping Theorem......Page 538
The Closed Graph Theorem......Page 542
Additional Problems for Chapter 12......Page 544
The Basic Inequalities......Page 547
The p and Lp Spaces (1p< )......Page 551
The Spaces  and L......Page 554
Separability......Page 556
The Spaces 2 and L2......Page 558
Continuous Linear Functionals......Page 564
The Lp Spaces (0Relations......Page 570
The Banach Algebra L1(IR)......Page 573
Weak Sequential Convergence......Page 579
Closed Subspaces of the Lp Spaces......Page 581
Additional Problems for Chapter 13......Page 584
Hilbert Spaces......Page 586
Inner Products......Page 587
Convex Sets......Page 592
Continuous Linear Functionals......Page 595
Orthogonal Series......Page 597
Weak Sequential Convergence......Page 603
Compact Operators......Page 607
Projections......Page 611
Eigenvectors and Eigenvalues......Page 613
Spectral Decomposition......Page 618
Additional Problems for Chapter 14......Page 622
Fourier Series......Page 625
Notation and Terminology......Page 626
Dirichlet\'s Kernel......Page 631
Fejér\'s Kernel......Page 634
Convergence of the Cesàro Means......Page 638
The Fourier Coefficients......Page 642
Weierstrass Approximation Theorem......Page 644
Pointwise Convergence: Jordan\'s Test......Page 647
Pointwise Convergence: Dini\'s Test......Page 652
Pointwise Divergence......Page 654
Characterizations......Page 656
Fourier Series in Hilbert Space......Page 658
Riemann\'s Theorems......Page 661
Cantor\'s Uniqueness Theorem......Page 665
Additional Problems for Chapter 15......Page 668
Index......Page 669