Analysis I

Analysis I......Page 3
Preface......Page 5
Preface to the English translation......Page 7
Chapter I: Foundations......Page 8
Chapter II: Convergence......Page 10
Chapter III: Continuous Functions......Page 11
Chapter IV: Differentiation in One Variable......Page 13
Index......Page 14
Chapter I: Foundations......Page 15
1. Fundamentals of Logic......Page 17
Exercises......Page 21
Elementary Facts......Page 22
Complement, Intersection and Union......Page 23
Products......Page 24
Families of Sets......Page 26
Exercises......Page 27
3. Functions......Page 29
Simple Examples......Page 30
Commutative Diagrams......Page 31
Injections, Surjections and Bijections......Page 32
Inverse Functions......Page 33
Set Valued Functions......Page 34
Exercises......Page 35
Equivalence Relations......Page 36
Order Relations......Page 37
Operations......Page 40
Exercises......Page 41
The Peano Axioms......Page 43
The Arithmetic of Natural Numbers......Page 45
The Division Algorithm......Page 48
The Induction Principle......Page 49
Recursive Definitions......Page 53
Exercises......Page 57
6. Countability......Page 60
Equinumerous Sets......Page 61
Countable Sets......Page 62
Infinite Products......Page 63
Exercises......Page 64
Groups......Page 66
Subgroups......Page 68
Cosets......Page 69
Homomorphisms......Page 70
Isomorphisms......Page 72
Exercises......Page 74
Rings......Page 76
The Multinomial Theorem......Page 79
Fields......Page 81
Ordered Fields......Page 83
Formal Power Series......Page 85
Polynomials......Page 87
Polynomial Functions......Page 89
Division of Polynomials......Page 90
Linear Factors......Page 91
Polynomials in Several Indeterminates......Page 92
Exercises......Page 94
The Integers......Page 98
The Rational Numbers......Page 99
Square Roots......Page 102
Exercises......Page 103
Order Completeness......Page 105
Dedekind’s Construction of the Real Numbers......Page 106
The Extended Number Line......Page 108
A Characterization of Supremum and Infimum......Page 109
The Density of the Rational Numbers in R......Page 110
Roots......Page 111
The Density of the Irrational Numbers in R......Page 113
Exercises......Page 114
Constructing the Complex Numbers......Page 117
Elementary Properties......Page 118
Computation with Complex Numbers......Page 120
Balls in K......Page 122
Exercises......Page 123
Vector Spaces......Page 125
Linear Functions......Page 126
Vector Space Bases......Page 129
Affine Spaces......Page 131
Affine Functions......Page 133
Polynomial Interpolation......Page 134
Algebras......Page 136
Difference Operators and Summation Formulas......Page 137
Newton Interpolation Polynomials......Page 138
Exercises......Page 140
Chapter II: Convergence......Page 143
Sequences......Page 145
Metric Spaces......Page 146
Cluster Points......Page 148
Convergence......Page 149
Uniqueness of the Limit......Page 151
Subsequences......Page 152
Exercises......Page 153
Elementary Rules......Page 155
The Comparison Test......Page 157
Complex Sequences......Page 158
Exercises......Page 160
Norms......Page 162
Balls......Page 163
Examples......Page 164
The Space of Bounded Functions......Page 165
Inner Product Spaces......Page 167
The Cauchy-Schwarz Inequality......Page 168
Euclidean Spaces......Page 170
Equivalent Norms......Page 171
Convergence in Product Spaces......Page 173
Exercises......Page 174
Bounded Monotone Sequences......Page 177
Some Important Limits......Page 178
Exercises......Page 181
Convergence to ±∞......Page 183
The Limit Superior and Limit Inferior......Page 184
The Bolzano-Weierstrass Theorem......Page 186
Exercises......Page 187
Cauchy Sequences......Page 189
Banach Spaces......Page 190
Cantor’s Construction of the Real Numbers......Page 191
Exercises......Page 195
Convergence of Series......Page 197
Harmonic and Geometric Series......Page 198
Convergence Tests......Page 199
Alternating Series......Page 200
Decimal, Binary and Other Representations of Real Numbers......Page 201
Exercises......Page 206
8. Absolute Convergence......Page 209
Majorant, Root and Ratio Tests......Page 210
Rearrangements of Series......Page 213
Double Series......Page 215
Cauchy Products......Page 218
Exercises......Page 221
9. Power Series......Page 224
The Radius of Convergence......Page 225
Addition and Multiplication of Power Series......Page 227
The Uniqueness of Power Series Representations......Page 228
Exercises......Page 229
Chapter III: Continuous Functions......Page 231
Elementary Properties and Examples......Page 233
Addition and Multiplication of Continuous Functions......Page 238
One-Sided Continuity......Page 242
Exercises......Page 243
Open Sets......Page 246
Closed Sets......Page 247
The Closure of a Set......Page 249
The Interior of a Set......Page 250
The Hausdorff Condition......Page 251
Examples......Page 252
A Characterization of Continuous Functions......Page 253
Continuous Extensions......Page 255
Relative Topology......Page 258
General Topological Spaces......Page 259
Exercises......Page 261
Covers......Page 264
A Characterization of Compact Sets......Page 265
Continuous Functions on Compact Spaces......Page 266
The Extreme Value Theorem......Page 267
Total Boundedness......Page 270
Uniform Continuity......Page 272
Compactness in General Topological Spaces......Page 273
Exercises......Page 274
Definition and Basic Properties......Page 277
Connectivity in R......Page 278
Path Connectivity......Page 279
Exercises......Page 282
Bolzano’s Intermediate Value Theorem......Page 285
Monotone Functions......Page 286
Continuous Monotone Functions......Page 288
Exercises......Page 289
Euler’s Formula......Page 291
The Real Exponential Function......Page 294
The Logarithm and Power Functions......Page 295
The Exponential Function on iR......Page 297
The Definition  of π and its Consequences......Page 299
The Tangent and Cotangent Functions......Page 303
The Complex Exponential Function......Page 304
Polar Coordinates......Page 305
Complex Logarithms......Page 307
Complex Powers......Page 308
A Further Representation of the Exponential Function......Page 309
Exercises......Page 310
Chapter IV: Differentiation in One Variable......Page 313
The Derivative......Page 315
Linear Approximation......Page 316
Rules for Differentiation......Page 318
The Chain Rule......Page 319
Inverse Functions......Page 320
Higher Derivatives......Page 321
One-Sided Differentiability......Page 327
Exercises......Page 329
Extrema......Page 331
The Mean Value Theorem......Page 332
Monotonicity and Differentiability......Page 333
Convexity and Differentiability......Page 336
The Inequalities of Young, Holder and Minkowski......Page 339
The Mean Value Theorem for Vector Valued Functions......Page 342
The Second Mean Value Theorem......Page 343
L’Hospital’s Rule......Page 344
Exercises......Page 345
The Landau Symbol......Page 349
Taylor’s Formula......Page 350
Taylor Polynomials and Taylor Series......Page 352
The Remainder Function in the Real Case......Page 354
Polynomial Interpolation......Page 358
Higher Order Difference Quotients......Page 359
Exercises......Page 361
Fixed Points and Contractions......Page 364
The Banach Fixed Point Theorem......Page 365
Newton’s Method......Page 369
Exercises......Page 372
Chapter V: Sequences of Functions......Page 374
Pointwise Convergence......Page 376
Uniform Convergence......Page 377
Series of Functions......Page 379
The Weierstrass Majorant Criterion......Page 380
Exercises......Page 382
Locally Uniform Convergence......Page 383
The Banach Space of Bounded Continuous Functions......Page 385
Differentiability......Page 386
Exercises......Page 387
Differentiability of Power Series......Page 390
Analyticity......Page 391
Antiderivatives of Analytic Functions......Page 393
The Power Series Expansion of the Logarithm......Page 394
The Binomial Series......Page 395
The Identity Theorem for Analytic Functions......Page 399
Exercises......Page 401
Banach Algebras......Page 403
Density and Separability......Page 404
The Stone-Weierstrass Theorem......Page 406
Trigonometric Polynomials......Page 409
Periodic Functions......Page 411
The Trigonometric Approximation Theorem......Page 414
Exercises......Page 415
Introduction to Mathematical Logic......Page 417
Bibliography......Page 422
Index......Page 423
Sachverzeichnis......Page 0