分析 第1卷 英文版 影印本PDF电子书下载

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