Analysis: an introduction

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
1A. Notation and Motivation......Page 13
1B. The Algebra of Various Number Systems......Page 17
1C. The Line and Cuts......Page 21
1D. Proofs, Generalizations, Abstractions, and Purposes......Page 24
2A. The Real Numbers......Page 27
2B. Decimal and Other Expansions; Countability......Page 33
2C. Algebraic and Transcendental Numbers......Page 36
2D. The Complex Numbers......Page 38
3A. Boundedness and Convergence......Page 42
3B. Upper and Lower Limits......Page 45
3C. The Cauchy Criterion......Page 47
3D. Algebraic Properties of Limits......Page 49
3E. Subsequences......Page 51
3F. The Extended Reals and Convergence to ±∞......Page 52
3G. Sizes of Things: The Logarithm......Page 54
4A. Convergence and Absolute Convergence......Page 57
4B. Tests for (Absolute) Convergence......Page 60
4C. Conditional Convergence......Page 66
4D. Euler’s Constant and Summation......Page 69
4E. Conditional Convergence: Summation by Parts......Page 70
5A. Power Series, Radius of Convergence......Page 73
5B. Differentiation of Power Series......Page 75
5C. Products and the Exponential Function......Page 78
5D. Abel’s Theorem and Summation......Page 82
6A. Metrics......Page 85
6B. Interior Points, Limit Points, Open and Closed Sets......Page 87
6C. Coverings and Compactness......Page 91
6D. Sequences, Completeness, Sequential Compactness......Page 93
6E. The Cantor Set......Page 96
7A. Definitions and General Properties......Page 98
7B. Real- and Complex-Valued Functions......Page 102
7C. The Space C(I)......Page 103
7D. Proof of the Weierstrass Polynomial Approximation Theorem......Page 107
8A. Differential Calculus......Page 111
8B. Inverse Functions......Page 117
8C. Integral Calculus......Page 119
8D. Riemann Sums......Page 124
8E. Two Versions of Taylor’s Theorem......Page 125
9A. The Complex Exponential Function and Related Functions......Page 131
9B. The Fundamental Theorem of Algebra......Page 136
9C. Infinite Products and Euler’s Formula for Sine......Page 137
10A. Introduction......Page 143
10B. Outer Measure......Page 145
10C. Measurable Sets......Page 148
10D. Fundamental Properties of Measurable Sets......Page 151
10E. A Nonmeasurable Set......Page 154
11A. Measurable Functions......Page 156
11B. Two Examples......Page 160
11C. Integration: Simple Functions......Page 161
11D. Integration: Measurable Functions......Page 163
11E. Convergence Theorems......Page 167
12A. Null Sets and the Notion of “Almost Everywhere”......Page 170
12B. Riemann Integration and Lebesgue Integration......Page 171
12C. The Space L1......Page 174
12D. The Space L2......Page 178
12E. Differentiating the Integral......Page 180
13A. Periodic Functions and Fourier Expansions......Page 185
13B. Fourier Coefficients of Integrable and Square-Integrable Periodic Functions......Page 188
13C. Dirichlet’s Theorem......Page 192
13D. Fejér\'s Theorem......Page 196
13E. The Weierstrass Approximation Theorem......Page 199
13F. L2-Periodic Functions: The Riesz-Fischer Theorem......Page 201
13G. More Convergence......Page 204
13H. Convolution......Page 207
14A. The Gibbs Phenomenon......Page 209
14B. A Continuous, Nowhere Differentiable Function......Page 211
14C. The Isoperimetric Inequality......Page 212
14D. Weyl’s Equidistribution Theorem......Page 214
14E. Strings......Page 215
14F. Woodwinds......Page 219
14G. Signals and the Fast Fourier Transform......Page 221
14H. The Fourier Integral......Page 223
14I. Position, Momentum, and the Uncertainty Principle......Page 227
15A. Introduction......Page 230
15B. Homogeneous Linear Equations......Page 231
15C. Constant Coefficient First-Order Systems......Page 235
15D. Nonuniqueness and Existence......Page 239
15E. Existence and Uniqueness......Page 242
15F. Linear Equations and Systems, Revisited......Page 246
Appendix The Banach-Tarski Paradox......Page 249
Section 2A......Page 253
Section 3A......Page 254
Section 3G......Page 255
Section 4C......Page 256
Section 6A......Page 257
Section 7A......Page 258
Section 8B......Page 259
Section 9A......Page 260
Section 10D......Page 261
Section 12D......Page 262
Section 13E......Page 263
Section 14D......Page 264
Section 15B......Page 265
Section 15F......Page 266
Notation Index......Page 267
General Index......Page 269