Distributions and Operators

Preface......Page 12
Introduction......Page 18
The Real Number System......Page 19
Algebraic Structure......Page 22
Order Structure......Page 25
Bounds......Page 26
Sups and Infs......Page 27
The Archimedean Property......Page 30
Inductive Property of IN......Page 32
The Rational Numbers Are Dense......Page 33
The Metric Structure of R......Page 35
Challenging Problems for Chapter 1......Page 38
Introduction......Page 40
Sequences......Page 42
Sequence Examples......Page 43
Countable Sets......Page 46
Convergence......Page 49
Divergence......Page 54
Boundedness Properties of Limits......Page 56
Algebra of Limits......Page 58
Order Properties of Limits......Page 64
Monotone Convergence Criterion......Page 69
Examples of Limits......Page 73
Subsequences......Page 78
Cauchy Convergence Criterion......Page 82
Upper and Lower Limits......Page 85
Challenging Problems for Chapter 2......Page 91
Introduction......Page 94
Finite Sums......Page 95
Infinite Unordered sums......Page 101
Cauchy Criterion......Page 103
Ordered Sums: Series......Page 107
Properties......Page 108
Special Series......Page 109
Criteria for Convergence......Page 115
Cauchy Criterion......Page 116
Absolute Convergence......Page 117
Trivial Test......Page 121
Direct Comparison Tests......Page 122
Limit Comparison Tests......Page 124
Ratio Comparison Test......Page 125
d\'Alembert\'s Ratio Test......Page 126
Cauchy\'s Root Test......Page 128
Cauchy\'s Condensation Test......Page 129
Integral Test......Page 131
Kummer\'s Tests......Page 132
Gauss\'s Ratio Test......Page 135
Alternating Series Test......Page 138
Dirichlet\'s Test......Page 139
Abel\'s Test......Page 140
Rearrangements......Page 146
Unconditional Convergence......Page 147
Conditional Convergence......Page 148
Comparison of i=1ai and iIN ai......Page 150
Products of Series......Page 152
Products of Absolutely Convergent Series......Page 155
Products of Nonabsolutely Convergent Series......Page 156
Summability Methods......Page 158
Cesàro\'s Method......Page 159
Abel\'s Method......Page 161
More on Infinite Sums......Page 165
Infinite Products......Page 167
Challenging Problems for Chapter 3......Page 171
Introduction......Page 175
Interior Points......Page 176
Points of Accumulation......Page 178
Boundary Points......Page 179
Sets......Page 182
Closed Sets......Page 183
Open Sets......Page 184
Elementary Topology......Page 190
Compactness Arguments......Page 193
Bolzano-Weierstrass Property......Page 195
Cantor\'s Intersection Property......Page 196
Cousin\'s Property......Page 198
Heine-Borel Property......Page 199
Compact Sets......Page 203
Countable Sets......Page 206
Challenging Problems for Chapter 4......Page 207
Limits (- Definition)......Page 210
Limits (Sequential Definition)......Page 214
Limits (Mapping Definition)......Page 217
One-Sided Limits......Page 218
Infinite Limits......Page 220
Properties of Limits......Page 221
Boundedness of Limits......Page 222
Algebra of Limits......Page 224
Order Properties......Page 227
Composition of Functions......Page 230
Examples......Page 232
Limits Superior and Inferior......Page 239
How to Define Continuity......Page 240
Continuity at a Point......Page 244
Continuity at an Arbitrary Point......Page 247
Continuity on a Set......Page 249
Properties of Continuous Functions......Page 252
Uniform Continuity......Page 253
Extremal Properties......Page 257
Darboux Property......Page 258
Types of Discontinuity......Page 260
Monotonic Functions......Page 262
How Many Points of Discontinuity?......Page 266
Challenging Problems for Chapter 5......Page 268
Dense Sets......Page 270
Nowhere Dense Sets......Page 272
A Two-Player Game......Page 274
The Baire Category Theorem......Page 276
Uniform Boundedness......Page 277
Construction of the Cantor Ternary Set......Page 279
An Arithmetic Construction of K......Page 282
The Cantor Function......Page 284
Sets of Type G......Page 286
Sets of Type F......Page 288
Oscillation and Continuity......Page 290
Oscillation of a Function......Page 291
The Set of Continuity Points......Page 294
Sets of Measure Zero......Page 296
Challenging Problems for Chapter 6......Page 302
The Derivative......Page 303
Definition of the Derivative......Page 304
Differentiability and Continuity......Page 309
The Derivative as a Magnification......Page 310
Computations of Derivatives......Page 311
Algebraic Rules......Page 312
The Chain Rule......Page 315
Inverse Functions......Page 319
The Power Rule......Page 320
Continuity of the Derivative?......Page 322
Local Extrema......Page 324
Mean Value Theorem......Page 326
Rolle\'s Theorem......Page 327
Mean Value Theorem......Page 329
Cauchy\'s Mean Value Theorem......Page 331
Monotonicity......Page 332
Dini Derivates......Page 335
The Darboux Property of the Derivative......Page 339
Convexity......Page 342
L\'Hôpital\'s Rule......Page 347
L\'Hôpital\'s Rule: 00 Form......Page 349
L\'Hôpital\'s Rule as x......Page 351
L\'Hôpital\'s Rule:  Form......Page 353
Taylor Polynomials......Page 356
Challenging Problems for Chapter 7......Page 360
Introduction......Page 363
Cauchy\'s First Method......Page 366
Scope of Cauchy\'s First Method......Page 368
Properties of the Integral......Page 371
Cauchy\'s Second Method......Page 376
Cauchy\'s Second Method (Continued)......Page 379
The Riemann Integral......Page 381
Some Examples......Page 383
Riemann\'s Criteria......Page 385
Lebesgue\'s Criterion......Page 387
What Functions Are Riemann Integrable?......Page 390
Properties of the Riemann Integral......Page 391
The Improper Riemann Integral......Page 395
More on the Fundamental Theorem of Calculus......Page 397
Challenging Problems for Chapter 8......Page 399
Introduction......Page 401
Pointwise Limits......Page 402
Uniform Limits......Page 408
The Cauchy Criterion......Page 411
Weierstrass M-Test......Page 413
Abel\'s Test for Uniform Convergence......Page 415
Uniform Convergence and Continuity......Page 421
Dini\'s Theorem......Page 422
Sequences of Continuous Functions......Page 425
Sequences of Riemann Integrable Functions......Page 427
Sequences of Improper Integrals......Page 429
Uniform Convergence and Derivatives......Page 432
Limits of Discontinuous Derivatives......Page 434
Pompeiu\'s Function......Page 436
Continuity and Pointwise Limits......Page 439
Challenging Problems for Chapter 9......Page 442
Introduction......Page 443
Power Series: Convergence......Page 444
Uniform Convergence......Page 449
Continuity of Power Series......Page 452
Integration of Power Series......Page 453
Differentiation of Power Series......Page 454
Power Series Representations......Page 457
The Taylor Series......Page 460
Representing a Function by a Taylor Series......Page 461
Analytic Functions......Page 464
Products of Power Series......Page 466
Quotients of Power Series......Page 467
Composition of Power Series......Page 469
Trigonometric Series......Page 470
Uniform Convergence of Trigonometric Series......Page 471
Fourier Series......Page 472
Convergence of Fourier Series......Page 473
Weierstrass Approximation Theorem......Page 477
The Algebraic Structure of Rn......Page 479
The Metric Structure of Rn......Page 481
Elementary Topology of Rn......Page 485
Sequences in Rn......Page 487
Functions from RnR......Page 492
Functions from RnRm......Page 494
Definition......Page 497
Coordinate-Wise Convergence......Page 500
Algebraic Properties......Page 502
Continuity of Functions from Rn to Rm......Page 503
Compact Sets in Rn......Page 506
Continuous Functions on Compact Sets......Page 507
Additional Remarks......Page 508
Introduction......Page 512
Partial and Directional Derivatives......Page 513
Partial Derivatives......Page 514
Directional Derivatives......Page 517
Cross Partials......Page 518
Integrals Depending on a Parameter......Page 523
Differentiable Functions......Page 527
Approximation by Linear Functions......Page 528
Definition of Differentiability......Page 529
Differentiability and Continuity......Page 533
Directional Derivatives......Page 534
An Example......Page 536
 Sufficient Conditions for Differentiability......Page 538
The Differential......Page 540
Preliminary Discussion......Page 543
Informal Proof of a Chain Rule......Page 547
Notation of Chain Rules......Page 548
Proofs of Chain Rules (I)......Page 550
Mean Value Theorem......Page 552
Proofs of Chain Rules (II)......Page 553
Higher Derivatives......Page 555
Implicit Function Theorems......Page 558
One-Variable Case......Page 559
Several-Variable Case......Page 562
Simultaneous Equations......Page 566
Inverse Function Theorem......Page 570
Functions From RRm......Page 573
Functions From RnRm......Page 576
Review of Differentials and Derivatives......Page 577
Definition of the Derivative......Page 579
Jacobians......Page 581
Chain Rules......Page 584
Proof of Chain Rule......Page 586
Introduction......Page 591
Metric Spaces---Specific Examples......Page 593
Sequence Spaces......Page 598
Function Spaces......Page 600
Convergence......Page 603
Sets in a Metric Space......Page 607
Functions......Page 615
Continuity......Page 617
Homeomorphisms......Page 622
Isometries......Page 628
Separable Spaces......Page 631
Complete Spaces......Page 634
Completeness Proofs......Page 635
Cantor Intersection Property......Page 637
Completion of a Metric Space......Page 638
Contraction Maps......Page 641
Applications of Contraction Maps (I)......Page 648
Applications of Contraction Maps (II)......Page 651
Systems of Equations (Example 13.79 Revisited)......Page 652
Infinite Systems (Example 13.80 revisited)......Page 653
Integral Equations (Example 13.81 revisited)......Page 655
Picard\'s Theorem (Example 13.82 revisited)......Page 656
Compactness......Page 658
The Bolzano-Weierstrass Property......Page 659
Continuous Functions on Compact Sets......Page 662
The Heine-Borel Property......Page 664
Total Boundedness......Page 666
Compact Sets in C[a,b]......Page 669
Peano\'s Theorem......Page 674
Baire Category Theorem......Page 677
Nowhere Dense Sets......Page 678
The Baire Category Theorem......Page 681
Functions Whose Graphs ``Cross No Lines\'\'......Page 684
Nowhere Monotonic Functions......Page 688
Continuous Nowhere Differentiable Functions......Page 689
Cantor Sets......Page 690
Challenging Problems for Chapter 13......Page 692
Set Notation......Page 696
Function Notation......Page 700
Why Proofs?......Page 706
Indirect Proof......Page 708
Contraposition......Page 709
Counterexamples......Page 710
Induction......Page 711
Quantifiers......Page 714
Appendix B: Hints for Selected Exercises......Page 717
SUBJECT INDEX......Page 734