Real Analysis (Bruckner and Thomson, 1997)

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      1 Background and Preview 1

        1.1 The Real Numbers 2
        1.2 Compact Sets of Real Numbers 7
        1.3 Countable Sets 10
        1.4 Uncountable Cardinals 13
        1.5 Transfinite Ordinals 16
        1.6 Category 19
        1.7 Outer Measure and Outer Content 22
        1.8 Small Sets 24
        1.9 Measurable Sets of Real Numbers 27
        1.10 Nonmeasurable Sets 31
        1.11 Zorn’s Lemma 34
        1.12 Borel Sets of Real Numbers 36
        1.13 Analytic Sets of Real Numbers 38
        1.14 Bounded Variation 40
        1.15 Newton’s Integral 43
        1.16 Cauchy’s Integral 44
        1.17 Riemann’s Integral 46
        1.18 Volterra’s Example 49
        1.19 Riemann–Stieltjes Integral 51
        1.20 Lebesgue’s Integral 54
        1.21 The Generalized Riemann Integral 56
        1.22 Additional Problems for Chapter 1 59
      2 Measure Spaces 63

        2.1 One-Dimensional Lebesgue Measure 64
        2.2 Additive Set Functions 69
        2.3 Measures and Signed Measures 75
        2.4 Limit Theorems 78
        2.5 Jordan and Hahn Decomposition 82
        2.6 Complete Measures 85
        2.7 Outer Measures 88
        2.8 Method I 92
        2.9 Regular Outer Measures 95
        2.10 Nonmeasurable Sets 99
        2.11 More About Method I 102
        2.12 Completions 105
        2.13 Additional Problems for Chapter 2 108
      2 Metric Outer Measures 112

        3.1 Metric Space 113
        3.2 Metric Outer Measures 116
        3.3 Method II 121
        3.4 Approximations 125
        3.5 Construction of Lebesgue–Stieltjes Measures 127
        3.6 Properties of Lebesgue–Stieltjes Measures 133
        3.7 Lebesgue–Stieltjes Measures in IRn 138
        3.8 HausdorffMeasures and HausdorffDimension 140
        3.9 Methods III and IV 147
        3.10 Additional Remarks 152
        3.11 Additional Problems for Chapter 3 156
      4 Measurable Functions 161

        4.1 Definitions and Basic Properties 162
        4.2 Sequences of Measurable Functions 167
        4.3 Egoroff’s Theorem 172
        4.4 Approximations by Simple Functions 175
        4.5 Approximation by Continuous Functions 179
        4.6 Additional Problems for Chapter 4 184
      5 Integration 188

        5.1 Introduction 189
        5.2 Integrals of Nonnegative Functions 193
        5.3 Fatou’s Lemma 197
        5.4 Integrable Functions 201
        5.5 Riemann and Lebesgue 205
        5.6 Countable Additivity of the Integral 213
        5.7 Absolute Continuity 216
        5.8 Radon–Nikodym Theorem 221
        5.9 Convergence Theorems 228
        5.10 Relations to Other Integrals 235
        5.11 Integration of Complex Functions 239
        5.12 Additional Problems for Chapter 5 243
      6 Fubini’s Theorem 248

        6.1 Product Measures 249
        6.2 Fubini’s Theorem 257
        6.3 Tonelli’s Theorem 259
        6.4 Additional Problems for Chapter 6 261
      7 Differentiation 264

        7.1 The Vitali Covering Theorem 264
        7.2 Functions of Bounded Variation 270
        7.3 The Banach–Zarecki Theorem 274
        7.4 Determining a Function by Its Derivative 277
        7.5 Calculating a Function from Its Derivative 279
        7.6 Total Variation of a Continuous Function 286
        7.7 VBG∗ Functions 292
        7.8 Approximate Continuity, Lebesgue Points 296
        7.9 Additional Problems for Chapter 7 302
      8 Differentiation of Measures 309

        8.1 Differentiation of Lebesgue–Stieltjes Measures 310
        8.2 The Cube Basis; Ordinary Differentiation 314
        8.3 The Lebesgue Decomposition Theorem 320
        8.4 The Interval Basis; Strong Differentiation 322
        8.5 Net Structures 329
        8.6 Radon–Nikodym Derivative in a Measure Space 335
        8.7 Summary, Comments, and References 343
        8.8 Additional Problems for Chapter 8 346
      9 Metric Spaces 348

        9.1 Definitions and Examples 348
        9.2 Convergence and Related Notions 357
        9.3 Continuity 360
        9.4 Homeomorphisms and Isometries 364
        9.5 Separable Spaces 368
        9.6 Complete Spaces 370
        9.7 Contraction Maps 375
        9.8 Applications of Contraction Mappings 377
        9.9 Compactness 383
        9.10 Totally Bounded Spaces 387
        9.11 Compact Sets in C(X) 388
        9.12 Application of the Arzel` a–Ascoli Theorem 392
        9.13 The Stone–Weierstrass Theorem 394
        9.14 The Isoperimetric Problem 397
        9.15 More on Convergence 400
        9.16 Additional Problems for Chapter 9 404
      10 Baire Category 407

        10.1 The Baire Category Theorem 407
        10.2 The Banach–Mazur Game 413
        10.3 The First Classes of Baire and Borel 418
        10.4 Properties of Baire-1 Functions 423
        10.5 Topologically Complete Spaces 427
        10.6 Applications to Function Spaces 431
        10.7 Additional Problems for Chapter 10 442

      11 Analytic Sets 448

        11.1 Products of Metric Spaces 449
        11.2 Baire Space 450
        11.3 Analytic Sets 453
        11.4 Borel Sets 457
        11.5 An Analytic Set That Is Not Borel 461
        11.6 Measurability of Analytic Sets 463
        11.7 The Suslin Operation 465
        11.8 A Method to Show a Set Is Not Borel 467
        11.9 Differentiable Functions 470
        11.10Additional Problems for Chapter 11 474
      12 Banach Spaces 477

        12.1 Normed Linear Spaces 477
        12.2 Compactness 483
        12.3 Linear Operators 487
        12.4 Banach Algebras 491
        12.5 The Hahn–Banach Theorem 494
        12.6 Improving Lebesgue Measure 498
        12.7 The Dual Space 504
        12.8 The Riesz Representation Theorem 507
        12.9 Separation of Convex Sets 513
        12.10An Embedding Theorem 518
        12.11The Uniform Boundedness Principle 520
        12.12An Application to Summability 523
        12.13The Open Mapping Theorem 527
        12.14The Closed Graph Theorem 531
        12.15Additional Problems for Chapter 12 533
      13 The Lp spaces 536

        13.1 The Basic Inequalities 536
        13.2 The p and Lp Spaces (1 ≤ p< ∞) 540
        13.3 The Spaces ∞ and L∞ 543
        13.4 Separability 545
        13.5 The Spaces 2 and L2 547
        13.6 Continuous Linear Functionals 553
        13.7 The Lp Spaces (0 <p< 1) 557
        13.8 Relations 559
        13.9 The Banach Algebra L1(IR) 562
        13.10Weak Sequential Convergence 568
        13.11Closed Subspaces of the Lp Spaces 570
        13.12Additional Problems for Chapter 13 573
      14 Hilbert Spaces 575

        14.1 Inner Products 576
        14.2 Convex Sets 581
        14.3 Continuous Linear Functionals 584
        14.4 Orthogonal Series 586
        14.5 Weak Sequential Convergence 592
        14.6 Compact Operators 596
        14.7 Projections 600
        14.8 Eigenvectors and Eigenvalues 602
        14.9 Spectral Decomposition 607
        14.10Additional Problems for Chapter 14 611
      15Fourier Series 614

        15.1 Notation and Terminology 615
        15.2 Dirichlet’s Kernel 620
        15.3 Fej´ er’s Kernel 623
        15.4 Convergence of the Ces` aro Means 627
        15.5 The Fourier Coefficients 631
        15.6 Weierstrass Approximation Theorem 633
        15.7 Pointwise Convergence: Jordan’s Test 636
        15.8 Pointwise Convergence: Dini’s Test 641
        15.9 Pointwise Divergence 643
        15.10Characterizations 645
        15.11Fourier Series in Hilbert Space 647
        15.12Riemann’s Theorems 650
        15.13Cantor’s Uniqueness Theorem 654
        15.14Additional Problems for Chapter 15 657

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