A Course in Mathematical Analysis - Vol 1: Foundations and Elementary Real Analysis

Symbolverzeichnis......Page 0
A COURSE IN MATHEMATICAL ANALYSIS......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Introduction......Page 17
Part I Functions of a real variable......Page 19
1.1 The need for axiomatic set theory......Page 21
1.2 The first few axioms of set theory......Page 23
1.3 Relations and partial orders......Page 27
1.4 Functions......Page 29
1.5 Equivalence relations......Page 34
1.6 Some theorems of set theory......Page 36
1.7 The foundation axiom and the axiom of infinity......Page 38
1.8 Sequences, and recursion......Page 41
1.9 The axiom of choice......Page 44
1.10 Concluding remarks......Page 47
2.1 The non-negative integers and the natural numbers......Page 50
2.2 Finite and infinite sets......Page 55
2.3 Countable sets......Page 60
2.4 Sequences and subsequences......Page 64
2.5 The integers......Page 67
2.6 Divisibility and factorization......Page 71
2.7 The field of rational numbers......Page 77
2.8 Ordered fields......Page 82
2.9 Dedekind cuts......Page 84
2.10 The real number field......Page 88
3.1 The real numbers......Page 97
3.2 Convergent sequences......Page 102
3.3 The uniqueness of the real number system......Page 109
3.4 The Bolzano-Weierstrass theorem......Page 112
3.5 Upper and lower limits......Page 113
3.6 The general principle of convergence......Page 116
3.7 Complex numbers......Page 117
3.8 The convergence of complex sequences......Page 123
4.1 Infinite series......Page 125
4.2 Series with non-negative terms......Page 127
4.3 Absolute and conditional convergence......Page 133
4.4 Iterated limits and iterated sums......Page 136
4.5 Rearranging series......Page 138
4.6 Convolution, or Cauchy, products......Page 141
4.7 Power series......Page 144
5.1 Closed sets......Page 149
5.2 Open sets......Page 153
5.3 Connectedness......Page 154
5.4 Compact sets......Page 156
5.5 Perfect sets, and Cantor's ternary set......Page 159
6.1 Limits and convergence of functions......Page 165
6.2 Orders of magnitude......Page 169
6.3 Continuity......Page 171
6.4 The intermediate value theorem......Page 180
6.5 Point-wise convergence and uniform convergence......Page 182
6.6 More on power series......Page 185
7.1 Differentiation at a point......Page 191
7.2 Convex functions......Page 198
7.3 Differentiable functions on an interval......Page 204
7.4 The exponential and logarithmic functions; powers......Page 207
7.5 The circular functions......Page 211
7.6 Higher derivatives, and Taylor's theorem......Page 218
8.1 Elementary integrals......Page 227
8.2 Upper and lower Riemann integrals......Page 229
8.3 Riemann integrable functions......Page 232
8.4 Algebraic properties of the Riemann integral......Page 238
8.5 The fundamental theorem of calculus......Page 241
8.6 Some mean-value theorems......Page 246
8.7 Integration by parts......Page 249
8.8 Improper integrals and singular integrals......Page 251
9.1 Introduction......Page 258
9.2 Complex Fourier series......Page 261
9.3 Uniqueness......Page 264
9.4 Convolutions, and Parseval's equation......Page 270
9.5 An example......Page 274
9.6 The Dirichlet kernel......Page 275
9.7 The Fejér kernel and the Poisson kernel......Page 282
10.1 Infinite products......Page 288
10.2 The Taylor series of logarithmic functions......Page 291
10.3 The beta function......Page 292
10.4 Stirling's formula......Page 295
10.5 The gamma function......Page 296
10.6 Riemann's zeta function......Page 299
10.7 Chebyshev's prime number theorem......Page 300
10.8 Evaluating ζ(2)......Page 304
10.9 The irrationality of er......Page 305
10.10 The irrationality of π......Page 307
A.1 Zorn's lemma......Page 309
A.2 The well-ordering principle......Page 311
Index......Page 313