Real Analysis and Foundations

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Preface to the Second Edition......Page 8
Overview......Page 10
How to Build a Course from this Text......Page 11
Audience......Page 12
Acknowledgements......Page 13
Table of Contents......Page 0
1.1 Introduction......Page 18
1.2 "And" and "Or"......Page 19
1.3 "Not" and "If-Then"......Page 21
1.4 Contrapositive, Converse, and "Iff......Page 24
1.5 Quantifiers......Page 27
1.6 Set Theory and Venn Diagrams......Page 30
1.7 Relations and Functions......Page 35
1.8 Countable and Uncountable Sets......Page 41
 EXERCISES......Page 51
2.1 The Natural Numbers......Page 56
2.2 Equivalence Relations and Equivalence Classes......Page 59
2.3 The Integers......Page 61
2.4 The Rational Numbers......Page 66
2.5 The Real Numbers......Page 67
2.6 The Complex Numbers......Page 79
EXERCISES......Page 84
3.1 Convergence of Sequences......Page 92
3.2 Subsequences......Page 98
3.3 Lim sup and Lim inf......Page 102
3 4 Some Special Sequences......Page 105
 EXERCISES......Page 108
4.1 Convergence of Series......Page 112
4.2 Elementary Convergence Tests......Page 117
4.3 Advanced Convergence Tests......Page 124
4.4 Some Special Series......Page 131
4.5 Operations on Series......Page 136
 EXERCISES......Page 139
5.1 Open and Closed Sets......Page 146
5.2 Further Properties of Open and Closed Sets......Page 151
5.3 Compact Sets......Page 156
5.4 The Cantor Set......Page 159
5.5 Connected and Disconnected Sets......Page 162
5.6 Perfect Sets......Page 164
 EXERCISES......Page 166
6.1 Definition and Basic Properties of the Limit of a Function......Page 170
6.2 Continuous Functions......Page 176
6.3 Topological Properties and Continuity......Page 181
6.4 Classifying Discontinuities and Monotonicity......Page 187
EXERCISES......Page 192
7.1 The Concept of Derivative......Page 198
7.2 The Mean Value Theorem and Applications......Page 206
7.3 More on the Theory of Differentiation......Page 214
EXERCISES......Page 218
8.1 Partitions and The Concept of integral......Page 222
8.2 Properties of the Riemann Integral......Page 228
8.3 Another Look at the Integral......Page 236
8.4 Advanced Results on Integration Theory......Page 241
 EXERCISES......Page 248
9.1 Partial Sums and Pointwise Convergence......Page 254
9.2 More on Uniform Convergence......Page 259
9.3 Series of Functions......Page 262
9.4 The Weierstrass Approximation Theorem......Page 265
EXERCISES......Page 269
10.1 Power Series......Page 274
10.2 More on Power Series: Convergence Issues......Page 279
10.3 The Exponential and Trigonometric Functions......Page 284
10.4 Logarithms and Powers of Real Numbers......Page 290
10.5 The Gamma Function and Stirling's Formula......Page 293
EXERCISES......Page 295
11.1.1 The Form of a Differential Equation......Page 302
11.1.2 Picard's Iteration Technique......Page 303
11.1.3 Some Illustrative Examples......Page 304
11.1.4 Estimation of the Picard Iterates......Page 306
11.2 The Method of Characteristics......Page 307
11.3 Power Series Methods......Page 310
EXERCISES......Page 318
12.1 The Idea of Harmonic Analysis......Page 324
12.2 The Elements of Fourier Series......Page 325
12.3 An Introduction to the Fourier Transform......Page 332
12.3.1 Appendix: Approximation by Smooth Functions......Page 336
12.4.1 Remarks on Different Fourier Notations......Page 341
12.4.2 The Dirichiet Problem on the Disc......Page 342
12.4.3 The Poisson Integral......Page 346
12.4.4 The Wave Equation......Page 348
EXERCISES......Page 353
13.1 Review of Linear Algebra......Page 362
13.2 A New Look at the Basic Concepts of Analysis......Page 368
13.3 Properties of the Derivative......Page 373
13.4 The Inverse and Implicit Function Theorems......Page 378
13.5 Differential Forms......Page 384
13.5.1 The Idea of a Differential Form......Page 385
13.5.2 Differential Forms on a Surface......Page 386
13.5.3 General Differential Forms and Stoker's Theorem......Page 389
EXERCISES......Page 392
14.1 Metric Spaces......Page 396
14.2 Topology in a Metric Space......Page 401
14.3 The Baire Category Theorem......Page 404
14.4 The Ascoli-Arzela Theorem......Page 408
14.5 The Lebesgue Integral......Page 411
14.5.1 Measurable Sets......Page 412
14.5.2 The Lebesgue Integral......Page 417
14.5.3 Calculating with the Lebesgue Integral......Page 420
14.6 A Taste of Probability Theory......Page 425
EXERCISES......Page 431
15.1 Localization in the Time and Space Variables......Page 438
15.2 A Custom Fourier Analysis......Page 441
15.3 The Haar Basis......Page 443
15.4 Some Illustrative Examples......Page 449
EXERCISES......Page 458
Bibliography......Page 462
Index......Page 464
Back Cover......Page 472