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分析 第1卷 英文版 影印本pdf电子书版本下载
- （德）阿莫恩（HERBERT AMANN），JOACHIM ESCHER著 著
- 出版社： 世界图书出版公司北京公司
- ISBN：7510047985
- 出版时间：2012
- 标注页数：426页
- 文件大小：65MB
- 文件页数：453页
- 主题词：
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图书目录
Chapter Ⅰ Foundations 3
1 Fundamentals of Logic 3
2 Sets 8
Elementary Facts 8
The Power Set 9
Complement,Intersection and Union 9
Products 10
Families of Sets 12
3 Functions 15
Simple Examples 16
Composition of Functions 17
Commutative Diagrams 17
Injections,Surjections and Bijections 18
Inverse Functions 19
Set Valued Functions 20
4 Relations and Operations 22
Equivalence Relations 22
Order Relations 23
Operations 26
5 The Natural Numbers 29
The Peano Axioms 29
The Arithmetic of Natural Numbers 31
The Division Algorithm 34
The Induction Principle 35
Recursive Definitions 39
6 Countability 46
Permutations 47
Equinumerous Sets 47
Countable Sets 48
Infinite Products 49
7 Groups and Homomorphisms 52
Groups 52
Subgroups 54
Cosets 55
Homomorphisms 56
Isomorphisms 58
8 Rings,Fields and Polynomials 62
Rings 62
The Binomial Theorem 65
The Multinomial Theorem 65
Fields 67
Ordered Fields 69
Formal Power Series 71
Polynomials 73
Polynomial Functions 75
Division of Polynomials 76
Linear Factors 77
Polynomials in Several Indeterminates 78
9 The Rational Numbers 84
The Integers 84
The Rational Numbers 85
Rational Zeros of Polynomials 88
Square Roots 88
10 The Real Numbers 91
Order Completeness 91
Dedekind's Construction of the Real Numbers 92
The Natural Order on R 94
The Extended Number Line 94
A Characterization of Supremum and Infimum 95
The Archimedean Property 96
The Density of the Rational Numbers in R 96
nth Roots 97
The Density of the Irrational Numbers in R 99
Intervals 100
11 The Complex Numbers 103
Constructing the Complex Numbers 103
Elementary Properties 104
Computation with Complex Numbers 106
Balls in K 108
12 Vector Spaces,Affine Spaces and Algebras 111
Vector Spaces 111
Linear Functions 112
Vector Space Bases 115
Affine Spaces 117
Affine Functions 119
Polynomial Interpolation 120
Algebras 122
Difference Operators and Summation Formulas 123
Newton Interpolation Polynomials 124
Chapter Ⅱ Convergence 131
1 Convergence of Sequences 131
Sequences 131
Metric Spaces 132
Cluster Points 134
Convergence 135
Bounded Sets 137
Uniqueness of the Limit 137
Subsequences 138
2 Real and Complex Sequences 141
Null Sequences 141
Elementary Rules 141
The Comparison Test 143
Complex Sequences 144
3 Normed Vector Spaces 148
Norms 148
Balls 149
Bounded Sets 150
Examples 150
The Space of Bounded Functions 151
Inner Product Spaces 153
The Cauchy-Schwarz Inequality 154
Euclidean Spaces 156
Equivalent Norms 157
Convergence in Product Spaces 159
4 Monotone Sequences 163
Bounded Monotone Sequences 163
Some Important Limits 164
5 Infinite Limits 169
Convergence to ±∞ 169
The Limit Superior and Limit Inferior 170
The Bolzano-Weierstrass Theorem 172
6 Completeness 175
Cauchy Sequences 175
Banach Spaces 176
Cantor's Construction of the Real Numbers 177
7 Series 183
Convergence of Series 183
Harmonic and Geometric Series 184
Calculating with Series 185
Convergence Tests 185
Alternating Series 186
Decimal,Binary and Other Representations of Real Numbers 187
The Uncountability of R 192
8 Absolute Convergence 195
Majorant,Root and Ratio Tests 196
The Exponential Function 199
Rearrangements of Series 199
Double Series 201
Cauchy Products 204
9 Power Series 210
The Radius of Convergence 211
Addition and Multiplication of Power Series 213
The Uniqueness of Power Series Representations 214
Chapter Ⅲ Continuous Functions 219
1 Continuity 219
Elementary Properties and Examples 219
Sequential Continuity 224
Addition and Multiplication of Continuous Functions 224
One-Sided Continuity 228
2 The Fundamentals of Topology 232
Open Sets 232
Closed Sets 233
The Closure of a Set 235
The Interior of a Set 236
The Boundary of a Set 237
The Hausdorff Condition 237
Examples 238
A Characterization of Continuous Functions 239
Continuous Extensions 241
Relative Topology 244
General Topological Spaces 245
3 Compactness 250
Covers 250
A Characterization of Compact Sets 251
Sequential Compactness 252
Continuous Functions on Compact Spaces 252
The Extreme Value Theorem 253
Total Boundedness 256
Uniform Continuity 258
Compactness in General Topological Spaces 259
4 Connectivity 263
Definition and Basic Properties 263
Connectivity in R 264
The Generalized Intermediate Value Theorem 265
Path Connectivity 265
Connectivity in General Topological Spaces 268
5 Functions on R 271
Bolzano's Intermediate Value Theorem 271
Monotone Functions 272
Continuous Monotone Functions 274
6 The Exponential and Related Functions 277
Euler's Formula 277
The Real Exponential Function 280
The Logarithm and Power Functions 281
The Exponential Function on iR 283
The Definition of π and its Consequences 285
The Tangent and Cotangent Functions 289
The Complex Exponential Function 290
Polar Coordinates 291
Complex Logarithms 293
Complex Powers 294
A Further Representation of the Exponential Function 295
Chapter Ⅳ Differentiation in One Variable 301
1 Differentiability 301
The Derivative 301
Linear Approximation 302
Rules for Difierentiation 304
The Chain Rule 305
Inverse Functions 306
Difierentiable Functions 307
Higher Derivatives 307
One-Sided Differentiability 313
2 The Mean Value Theorem and its Applications 317
Extrema 317
The Mean Value Theorem 318
Monotonicity and Differentiability 319
Convexity and Differentiability 322
The Inequalities of Young,H？lder and Minkowski 325
The Mean Value Theorem for Vector Valued Functions 328
The Second Mean Value Theorem 329
L'Hospital's Rule 330
3 Taylor's Theorem 335
The Landau Symbol 335
Taylor's Formula 336
Taylor Polynomials and Taylor Series 338
The Remainder Function in the Real Case 340
Polynomial Interpolation 344
Higher Order Difference Quotients 345
4 Iterative Procedures 350
Fixed Points and Contractions 350
The Banach Fixed Point Theorem 351
Newton's Method 355
Chapter Ⅴ Sequences of Functions 363
1 Uniform Convergence 363
Pointwise Convergence 363
Uniform Convergence 364
Series of Functions 366
The Weierstrass Majorant Criterion 367
2 Continuity and Differentiability for Sequences of Functions 370
Continuity 370
Locally Uniform Convergence 370
The Banach Space of Bounded Continuous Functions 372
Differentiability 373
3 Analytic Functions 377
Differentiability of Power Series 377
Analyticity 378
Antiderivatives of Analytic Functions 380
The Power Series Expansion of the Logarithm 381
The Binomial Series 382
The Identity Theorem for Analytic Functions 386
4 Polynomial Approximation 390
Banach Algebras 390
Density and Separability 391
The Stone-Weierstrass Theorem 393
Trigonometric Polynomials 396
Periodic Functions 398
The Trigonometric Approximation Theorem 401
Appendix Introduction to Mathematical Logic 405
Bibliography 411
Index 413