Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2)

Cover......Page 1
Series......Page 3
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Volume 1......Page 8
Volume 2......Page 11
Preface......Page 14
1.1 What is analysis?......Page 20
1.2 Why do analysis?......Page 22
2 – Starting at the beginning: the natural numbers......Page 33
2.1 The Peano axioms......Page 35
2.2 Addition......Page 46
2.3 Multiplication......Page 52
3.1 Fundamentals......Page 56
3.2 Russell’s paradox (Optional)......Page 71
3.3 Functions......Page 74
3.4 Images and inverse images......Page 83
3.5 Cartesian products......Page 89
3.6 Cardinality of sets......Page 95
4.1 The integers......Page 103
4.2 The rationals......Page 111
4.3 Absolute value and exponentiation......Page 117
4.4 Gaps in the rational numbers......Page 122
5 – The real numbers......Page 126
5.1 Cauchy sequences......Page 128
5.2 Equivalent Cauchy sequences......Page 133
5.3 The construction of the real numbers......Page 136
5.4 Ordering the reals......Page 146
5.5 The least upper bound property......Page 152
5.6 Real exponentiation, part I......Page 158
6.1 Convergence and limit laws......Page 164
6.2 The extended real number system......Page 172
6.3 Suprema and infima of sequences......Page 176
6.4 Limsup, liminf, and limit points......Page 179
6.5 Some standard limits......Page 189
6.6 Subsequences......Page 190
6. 7 Real exponentiation, part II......Page 194
7.1 Finite series......Page 198
7.2 Infinite series......Page 208
7.3 Sums of non-negative numbers......Page 214
7.4 Rearrangement of series......Page 219
7.5 The root and ratio tests......Page 223
8.1 Countability......Page 227
8.2 Summation on infinite sets......Page 235
8.3 Uncountable sets......Page 243
8.4 The axiom of choice......Page 246
8.5 Ordered sets......Page 251
9 – Continuous functions on R......Page 261
9.1 Subsets of the real line......Page 262
9.2 The algebra of real-valued functions......Page 269
9.3 Limiting values of functions......Page 272
9.4 Continuous functions......Page 280
9.5 Left and right limits......Page 285
9.6 The maximum principle......Page 288
9. 7 The intermediate value theorem......Page 292
9.8 Monotonic functions......Page 295
9.9 Uniform continuity......Page 298
9.10 Limits at infinity......Page 305
10.1 Basic definitions......Page 307
10.2 Local maxima, local minima, and derivatives......Page 314
10.3 Monotone functions and derivatives......Page 317
10.4 Inverse functions and derivatives......Page 319
10.5 L’Hôpital’s rule......Page 322
11 – The Riemann integral......Page 325
11.1 Partitions......Page 326
11.2 Piecewise constant functions......Page 331
11.3 Upper and lower Riemann integrals......Page 336
11.4 Basic properties of the Riemann integral......Page 340
11.5 Riemann integrability of continuous functions......Page 345
11.6 Riemann integrability of monotone functions......Page 349
11.7 A non-Riemann integrable function......Page 351
11.8 The Riemann-Stieltjes integral......Page 353
11.9 The two fundamental theorems of calculus......Page 357
11.10 Consequences of the fundamental theorems......Page 362
Appendix A: the basics of mathematical logic......Page 368
A.1 Mathematical statements......Page 369
A.2 Implication......Page 376
A.3 The structure of proofs......Page 383
A.4 Variables and quantifiers......Page 386
A.5 Nested quantifiers......Page 391
A.6 Some examples of proofs and quantifiers......Page 394
A.7 Equality......Page 396
Appendix B: the decimal system......Page 399
B.1 The decimal representation of natural numbers......Page 400
B.2 The decimal representation of real numbers......Page 404
A......Page 408
B......Page 409
C......Page 410
D......Page 411
E......Page 412
F-G-H-I......Page 413
J-L......Page 414
M-N......Page 416
O-P......Page 417
Q-R......Page 418
S......Page 419
T-U......Page 420
V-W-Z......Page 421
Cover......Page 422
Series......Page 424
Title......Page 425
Copyright......Page 426
Dedication......Page 427
Volume 1......Page 429
Volume 2......Page 432
Preface......Page 435
12.1 Definitions and examples......Page 441
12.2 Some point-set topology of metric spaces......Page 452
12.3 Relative topology......Page 457
12.4 Cauchy sequences and complete metric spaces......Page 460
12.5 Compact metric spaces......Page 464
13.1 Continuous functions......Page 472
13.2 Continuity and product spaces......Page 475
13.3 Continuity and compactness......Page 479
13.4 Continuity and connectedness......Page 481
13.5 Topological spaces (Optional)......Page 485
14 – Uniform convergence......Page 492
14.1 Limiting values of functions......Page 493
14.2 Pointwise and uniform convergence......Page 496
14.3 Uniform convergence and continuity......Page 501
14.4 The metric of uniform convergence......Page 504
14.5 Series of functions, the Weierstrass M-test......Page 507
14.6 Uniform convergence and integration......Page 510
14.7 Uniform convergence and derivatives......Page 513
14.8 Uniform approximation by polynomials......Page 516
15.1 Formal power series......Page 526
15.2 Real analytic functions......Page 529
15.3 Abel's theorem......Page 535
15.4 Multiplication of power series......Page 539
15.5 The exponential and logarithm functions......Page 542
15.6 A digression on complex numbers......Page 546
15.7 Trigonometric functions......Page 555
16 – Fourier series......Page 562
16.1 Periodic functions......Page 563
16.2 Inner products on periodic functions......Page 566
16.3 Trigonometric polynomials......Page 570
16.4 Periodic convolutions......Page 573
16.5 The Fourier and Plancherel theorems......Page 578
17.1 Linear transformations......Page 585
17.2 Derivatives in several variable calculus......Page 592
17.3 Partial and directional derivatives......Page 596
17.4 The several variable calculus chain rule......Page 604
17.5 Double derivatives and Clairaut's theorem......Page 607
17.6 The contraction mapping theorem......Page 610
17.7 The inverse function theorem......Page 613
17.8 The implicit function theorem......Page 619
18 – Lebesgue measure......Page 625
18.1 The goal: Lebesgue measure......Page 627
18.2 First attempt: Outer measure......Page 629
18.3 Outer measure is not additive......Page 639
18.4 Measurable sets......Page 642
18.5 Measurable functions......Page 649
19.1 Simple functions......Page 654
19.2 Integration of non-negative measurable functions......Page 660
19.3 Integration of absolutely integrable functions......Page 669
19.4 Comparison with the Riemann integral......Page 674
19.5 Fubini's theorem......Page 676
A......Page 683
B......Page 684
C......Page 685
D......Page 686
E......Page 687
F-G-H-I......Page 688
J-L......Page 689
M-N......Page 691
O-P......Page 692
Q-R......Page 693
S......Page 694
T-U......Page 695
V-W-Z......Page 696
Series “Texts and Readings in Mathematics”......Page 697