分析 第1卷 英文版 影印本PDF|Epub|txt|kindle电子书版本网盘下载

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Chapter Ⅰ Foundations3

1 Fundamentals of Logic3

2 Sets8

Elementary Facts8

The Power Set9

Complement,Intersection and Union9

Products10

Families of Sets12

3 Functions15

Simple Examples16

Composition of Functions17

Commutative Diagrams17

Injections,Surjections and Bijections18

Inverse Functions19

Set Valued Functions20

4 Relations and Operations22

Equivalence Relations22

Order Relations23

Operations26

5 The Natural Numbers29

The Peano Axioms29

The Arithmetic of Natural Numbers31

The Division Algorithm34

The Induction Principle35

Recursive Definitions39

6 Countability46

Permutations47

Equinumerous Sets47

Countable Sets48

Infinite Products49

7 Groups and Homomorphisms52

Groups52

Subgroups54

Cosets55

Homomorphisms56

Isomorphisms58

8 Rings,Fields and Polynomials62

Rings62

The Binomial Theorem65

The Multinomial Theorem65

Fields67

Ordered Fields69

Formal Power Series71

Polynomials73

Polynomial Functions75

Division of Polynomials76

Linear Factors77

Polynomials in Several Indeterminates78

9 The Rational Numbers84

The Integers84

The Rational Numbers85

Rational Zeros of Polynomials88

Square Roots88

10 The Real Numbers91

Order Completeness91

Dedekind's Construction of the Real Numbers92

The Natural Order on R94

The Extended Number Line94

A Characterization of Supremum and Infimum95

The Archimedean Property96

The Density of the Rational Numbers in R96

nth Roots97

The Density of the Irrational Numbers in R99

Intervals100

11 The Complex Numbers103

Constructing the Complex Numbers103

Elementary Properties104

Computation with Complex Numbers106

Balls in K108

12 Vector Spaces,Affine Spaces and Algebras111

Vector Spaces111

Linear Functions112

Vector Space Bases115

Affine Spaces117

Affine Functions119

Polynomial Interpolation120

Algebras122

Difference Operators and Summation Formulas123

Newton Interpolation Polynomials124

Chapter Ⅱ Convergence131

1 Convergence of Sequences131

Sequences131

Metric Spaces132

Cluster Points134

Convergence135

Bounded Sets137

Uniqueness of the Limit137

Subsequences138

2 Real and Complex Sequences141

Null Sequences141

Elementary Rules141

The Comparison Test143

Complex Sequences144

3 Normed Vector Spaces148

Norms148

Balls149

Bounded Sets150

Examples150

The Space of Bounded Functions151

Inner Product Spaces153

The Cauchy-Schwarz Inequality154

Euclidean Spaces156

Equivalent Norms157

Convergence in Product Spaces159

4 Monotone Sequences163

Bounded Monotone Sequences163

Some Important Limits164

5 Infinite Limits169

Convergence to ±∞169

The Limit Superior and Limit Inferior170

The Bolzano-Weierstrass Theorem172

6 Completeness175

Cauchy Sequences175

Banach Spaces176

Cantor's Construction of the Real Numbers177

7 Series183

Convergence of Series183

Harmonic and Geometric Series184

Calculating with Series185

Convergence Tests185

Alternating Series186

Decimal,Binary and Other Representations of Real Numbers187

The Uncountability of R192

8 Absolute Convergence195

Majorant,Root and Ratio Tests196

The Exponential Function199

Rearrangements of Series199

Double Series201

Cauchy Products204

9 Power Series210

The Radius of Convergence211

Addition and Multiplication of Power Series213

The Uniqueness of Power Series Representations214

Chapter Ⅲ Continuous Functions219

1 Continuity219

Elementary Properties and Examples219

Sequential Continuity224

Addition and Multiplication of Continuous Functions224

One-Sided Continuity228

2 The Fundamentals of Topology232

Open Sets232

Closed Sets233

The Closure of a Set235

The Interior of a Set236

The Boundary of a Set237

The Hausdorff Condition237

Examples238

A Characterization of Continuous Functions239

Continuous Extensions241

Relative Topology244

General Topological Spaces245

3 Compactness250

Covers250

A Characterization of Compact Sets251

Sequential Compactness252

Continuous Functions on Compact Spaces252

The Extreme Value Theorem253

Total Boundedness256

Uniform Continuity258

Compactness in General Topological Spaces259

4 Connectivity263

Definition and Basic Properties263

Connectivity in R264

The Generalized Intermediate Value Theorem265

Path Connectivity265

Connectivity in General Topological Spaces268

5 Functions on R271

Bolzano's Intermediate Value Theorem271

Monotone Functions272

Continuous Monotone Functions274

6 The Exponential and Related Functions277

Euler's Formula277

The Real Exponential Function280

The Logarithm and Power Functions281

The Exponential Function on iR283

The Definition of π and its Consequences285

The Tangent and Cotangent Functions289

The Complex Exponential Function290

Polar Coordinates291

Complex Logarithms293

Complex Powers294

A Further Representation of the Exponential Function295

Chapter Ⅳ Differentiation in One Variable301

1 Differentiability301

The Derivative301

Linear Approximation302

Rules for Difierentiation304

The Chain Rule305

Inverse Functions306

Difierentiable Functions307

Higher Derivatives307

One-Sided Differentiability313

2 The Mean Value Theorem and its Applications317

Extrema317

The Mean Value Theorem318

Monotonicity and Differentiability319

Convexity and Differentiability322

The Inequalities of Young,H？lder and Minkowski325

The Mean Value Theorem for Vector Valued Functions328

The Second Mean Value Theorem329

L'Hospital's Rule330

3 Taylor's Theorem335

The Landau Symbol335

Taylor's Formula336

Taylor Polynomials and Taylor Series338

The Remainder Function in the Real Case340

Polynomial Interpolation344

Higher Order Difference Quotients345

4 Iterative Procedures350

Fixed Points and Contractions350

The Banach Fixed Point Theorem351

Newton's Method355

Chapter Ⅴ Sequences of Functions363

1 Uniform Convergence363

Pointwise Convergence363

Uniform Convergence364

Series of Functions366

The Weierstrass Majorant Criterion367

2 Continuity and Differentiability for Sequences of Functions370

Continuity370

Locally Uniform Convergence370

The Banach Space of Bounded Continuous Functions372

Differentiability373

3 Analytic Functions377

Differentiability of Power Series377

Analyticity378

Antiderivatives of Analytic Functions380

The Power Series Expansion of the Logarithm381

The Binomial Series382

The Identity Theorem for Analytic Functions386

4 Polynomial Approximation390

Banach Algebras390

Density and Separability391

The Stone-Weierstrass Theorem393

Trigonometric Polynomials396

Periodic Functions398

The Trigonometric Approximation Theorem401

Appendix Introduction to Mathematical Logic405

Bibliography411

Index413