数学分析基本原理（影印版）
定价：￥169.00
作者： Paul J.Sally，Jr.
出版社：高等教育出版社
- 高等教育出版社
- 9787040659672
- 1版
- 64270644-4
- 特殊
- 理学
- 数学类
- 数学类
- 本科 研究生及以上
目录
目录
前辅文
Preface
Acknowledgments
Chapter 1. The Construction of Real and Complex Numbers
1.1. The Least Upper Bound Property and the Real Numbers
1.2. Consequences of the Least Upper Bound Property
1.3. Rational Approximation
1.4. Intervals
1.5. The Construction of the Real Numbers
1.6. Convergence in R
1.7. Automorphisms of Fields
1.8. Complex Numbers
1.9. Convergence in C
1.10. Independent Projects
Chapter 2. Metric and Euclidean Spaces
2.1. Introduction
2.2. Definition and Basic Properties of Metric Spaces
2.3. Topology of Metric Spaces
2.4. Limits and Continuous Functions
2.5. Absolute Continuity and Bounded Variation in R
2.6. Compactness, Completeness, and Connectedness
2.7. Independent Projects
Chapter 3. Complete Metric Spaces
3.1. The Contraction Mapping Theorem and Its Applications to Differential and Integral Equations
3.2. The Baire Category Theorem and the Uniform Boundedness Principle
3.3. Stone-Weierstrass Theorem
3.4. The p-adic Completion of Q
3.5. Independent Projects
Chapter 4. Normed Linear Spaces
4.1. Definitions and Basic Properties
4.2. Bounded Linear Operators
4.3. Fundamental Theorems about Linear Operators
4.4. Extending Linear Functionals
4.5. Generalized Limits and the Dual of l∞(F)
4.6. Adjoint Operators and Isometries of Normed Linear Spaces
4.7. Concrete Facts about Isometries of Normed Linear Spaces
4.8. Locally Compact Groups
4.9. Hilbert Spaces
4.10. Convergence and Selfadjoint Operators
4.11. Independent Projects
Chapter 5. Differentiation
5.1. Review of Differentiation in One Variable
5.2. Differential Calculus in Rn
5.3. The Derivative as a Matrix of Partial Derivatives
5.4. The Mean Value Theorem
5.5. Higher-Order Partial Derivatives and Taylor’s Theorem
5.6. Hypersurfaces and Tangent Hyperplanes in Rn
5.7. Max-Min Problems
5.8. Lagrange Multipliers
5.9. The Implicit and Inverse Function Theorems
5.10. Independent Projects
Chapter 6. Integration
6.1. Measures
6.2. Lebesgue Measure
6.3. Measurable Functions
6.4. The Integral
6.5. Lp Spaces
6.6. Fubini’s Theorem
6.7. Change of Variables in Integration
6.8. Independent Projects
Chapter 7. Fourier Analysis on Locally Compact Abelian Groups
7.1. Fourier Analysis on the Circle
7.2. Fourier Analysis on Locally Compact Abelian Groups
7.3. The Determination of ^G
7.4. The Fourier Transform on (R, +)
7.5. Fourier Inversion on (R, +)
7.6. Fourier Analysis on p-adic Fields
7.7. Independent Projects
Appendix A. Sets, Functions, and Other Basic Ideas
A.1. Sets and Elements
A.2. Equality, Inclusion, and Notation
A.3. The Algebra of Sets
A.4. Cartesian Products, Counting, and Power Sets
A.5. Some Sets of Numbers
A.6. Equivalence Relations and the Construction of Q
A.7. Functions
A.8. Countability and Other Basic Ideas
A.9. The Axiom of Choice
A.10. Independent Projects
Appendix B. Linear Algebra
B.1. Fundamentals of Linear Algebra
B.2. Linear Transformations
B.3. Linear Transformations and Matrices
B.4. Determinants
B.5. Geometric Linear Algebra
B.6. Independent Projects
Bibliography
Index of Terminology
Index of Notation Definitions
前辅文
Preface
Acknowledgments
Chapter 1. The Construction of Real and Complex Numbers
1.1. The Least Upper Bound Property and the Real Numbers
1.2. Consequences of the Least Upper Bound Property
1.3. Rational Approximation
1.4. Intervals
1.5. The Construction of the Real Numbers
1.6. Convergence in R
1.7. Automorphisms of Fields
1.8. Complex Numbers
1.9. Convergence in C
1.10. Independent Projects
Chapter 2. Metric and Euclidean Spaces
2.1. Introduction
2.2. Definition and Basic Properties of Metric Spaces
2.3. Topology of Metric Spaces
2.4. Limits and Continuous Functions
2.5. Absolute Continuity and Bounded Variation in R
2.6. Compactness, Completeness, and Connectedness
2.7. Independent Projects
Chapter 3. Complete Metric Spaces
3.1. The Contraction Mapping Theorem and Its Applications to Differential and Integral Equations
3.2. The Baire Category Theorem and the Uniform Boundedness Principle
3.3. Stone-Weierstrass Theorem
3.4. The p-adic Completion of Q
3.5. Independent Projects
Chapter 4. Normed Linear Spaces
4.1. Definitions and Basic Properties
4.2. Bounded Linear Operators
4.3. Fundamental Theorems about Linear Operators
4.4. Extending Linear Functionals
4.5. Generalized Limits and the Dual of l∞(F)
4.6. Adjoint Operators and Isometries of Normed Linear Spaces
4.7. Concrete Facts about Isometries of Normed Linear Spaces
4.8. Locally Compact Groups
4.9. Hilbert Spaces
4.10. Convergence and Selfadjoint Operators
4.11. Independent Projects
Chapter 5. Differentiation
5.1. Review of Differentiation in One Variable
5.2. Differential Calculus in Rn
5.3. The Derivative as a Matrix of Partial Derivatives
5.4. The Mean Value Theorem
5.5. Higher-Order Partial Derivatives and Taylor’s Theorem
5.6. Hypersurfaces and Tangent Hyperplanes in Rn
5.7. Max-Min Problems
5.8. Lagrange Multipliers
5.9. The Implicit and Inverse Function Theorems
5.10. Independent Projects
Chapter 6. Integration
6.1. Measures
6.2. Lebesgue Measure
6.3. Measurable Functions
6.4. The Integral
6.5. Lp Spaces
6.6. Fubini’s Theorem
6.7. Change of Variables in Integration
6.8. Independent Projects
Chapter 7. Fourier Analysis on Locally Compact Abelian Groups
7.1. Fourier Analysis on the Circle
7.2. Fourier Analysis on Locally Compact Abelian Groups
7.3. The Determination of ^G
7.4. The Fourier Transform on (R, +)
7.5. Fourier Inversion on (R, +)
7.6. Fourier Analysis on p-adic Fields
7.7. Independent Projects
Appendix A. Sets, Functions, and Other Basic Ideas
A.1. Sets and Elements
A.2. Equality, Inclusion, and Notation
A.3. The Algebra of Sets
A.4. Cartesian Products, Counting, and Power Sets
A.5. Some Sets of Numbers
A.6. Equivalence Relations and the Construction of Q
A.7. Functions
A.8. Countability and Other Basic Ideas
A.9. The Axiom of Choice
A.10. Independent Projects
Appendix B. Linear Algebra
B.1. Fundamentals of Linear Algebra
B.2. Linear Transformations
B.3. Linear Transformations and Matrices
B.4. Determinants
B.5. Geometric Linear Algebra
B.6. Independent Projects
Bibliography
Index of Terminology
Index of Notation Definitions
前辅文
Preface
Acknowledgments
Chapter 1. The Construction of Real and Complex Numbers
1.1. The Least Upper Bound Property and the Real Numbers
1.2. Consequences of the Least Upper Bound Property
1.3. Rational Approximation
1.4. Intervals
1.5. The Construction of the Real Numbers
1.6. Convergence in R
1.7. Automorphisms of Fields
1.8. Complex Numbers
1.9. Convergence in C
1.10. Independent Projects
Chapter 2. Metric and Euclidean Spaces
2.1. Introduction
2.2. Definition and Basic Properties of Metric Spaces
2.3. Topology of Metric Spaces
2.4. Limits and Continuous Functions
2.5. Absolute Continuity and Bounded Variation in R
2.6. Compactness, Completeness, and Connectedness
2.7. Independent Projects
Chapter 3. Complete Metric Spaces
3.1. The Contraction Mapping Theorem and Its Applications to Differential and Integral Equations
3.2. The Baire Category Theorem and the Uniform Boundedness Principle
3.3. Stone-Weierstrass Theorem
3.4. The p-adic Completion of Q
3.5. Independent Projects
Chapter 4. Normed Linear Spaces
4.1. Definitions and Basic Properties
4.2. Bounded Linear Operators
4.3. Fundamental Theorems about Linear Operators
4.4. Extending Linear Functionals
4.5. Generalized Limits and the Dual of l∞(F)
4.6. Adjoint Operators and Isometries of Normed Linear Spaces
4.7. Concrete Facts about Isometries of Normed Linear Spaces
4.8. Locally Compact Groups
4.9. Hilbert Spaces
4.10. Convergence and Selfadjoint Operators
4.11. Independent Projects
Chapter 5. Differentiation
5.1. Review of Differentiation in One Variable
5.2. Differential Calculus in Rn
5.3. The Derivative as a Matrix of Partial Derivatives
5.4. The Mean Value Theorem
5.5. Higher-Order Partial Derivatives and Taylor’s Theorem
5.6. Hypersurfaces and Tangent Hyperplanes in Rn
5.7. Max-Min Problems
5.8. Lagrange Multipliers
5.9. The Implicit and Inverse Function Theorems
5.10. Independent Projects
Chapter 6. Integration
6.1. Measures
6.2. Lebesgue Measure
6.3. Measurable Functions
6.4. The Integral
6.5. Lp Spaces
6.6. Fubini’s Theorem
6.7. Change of Variables in Integration
6.8. Independent Projects
Chapter 7. Fourier Analysis on Locally Compact Abelian Groups
7.1. Fourier Analysis on the Circle
7.2. Fourier Analysis on Locally Compact Abelian Groups
7.3. The Determination of ^G
7.4. The Fourier Transform on (R, +)
7.5. Fourier Inversion on (R, +)
7.6. Fourier Analysis on p-adic Fields
7.7. Independent Projects
Appendix A. Sets, Functions, and Other Basic Ideas
A.1. Sets and Elements
A.2. Equality, Inclusion, and Notation
A.3. The Algebra of Sets
A.4. Cartesian Products, Counting, and Power Sets
A.5. Some Sets of Numbers
A.6. Equivalence Relations and the Construction of Q
A.7. Functions
A.8. Countability and Other Basic Ideas
A.9. The Axiom of Choice
A.10. Independent Projects
Appendix B. Linear Algebra
B.1. Fundamentals of Linear Algebra
B.2. Linear Transformations
B.3. Linear Transformations and Matrices
B.4. Determinants
B.5. Geometric Linear Algebra
B.6. Independent Projects
Bibliography
Index of Terminology
Index of Notation Definitions
前辅文
Preface
Acknowledgments
Chapter 1. The Construction of Real and Complex Numbers
1.1. The Least Upper Bound Property and the Real Numbers
1.2. Consequences of the Least Upper Bound Property
1.3. Rational Approximation
1.4. Intervals
1.5. The Construction of the Real Numbers
1.6. Convergence in R
1.7. Automorphisms of Fields
1.8. Complex Numbers
1.9. Convergence in C
1.10. Independent Projects
Chapter 2. Metric and Euclidean Spaces
2.1. Introduction
2.2. Definition and Basic Properties of Metric Spaces
2.3. Topology of Metric Spaces
2.4. Limits and Continuous Functions
2.5. Absolute Continuity and Bounded Variation in R
2.6. Compactness, Completeness, and Connectedness
2.7. Independent Projects
Chapter 3. Complete Metric Spaces
3.1. The Contraction Mapping Theorem and Its Applications to Differential and Integral Equations
3.2. The Baire Category Theorem and the Uniform Boundedness Principle
3.3. Stone-Weierstrass Theorem
3.4. The p-adic Completion of Q
3.5. Independent Projects
Chapter 4. Normed Linear Spaces
4.1. Definitions and Basic Properties
4.2. Bounded Linear Operators
4.3. Fundamental Theorems about Linear Operators
4.4. Extending Linear Functionals
4.5. Generalized Limits and the Dual of l∞(F)
4.6. Adjoint Operators and Isometries of Normed Linear Spaces
4.7. Concrete Facts about Isometries of Normed Linear Spaces
4.8. Locally Compact Groups
4.9. Hilbert Spaces
4.10. Convergence and Selfadjoint Operators
4.11. Independent Projects
Chapter 5. Differentiation
5.1. Review of Differentiation in One Variable
5.2. Differential Calculus in Rn
5.3. The Derivative as a Matrix of Partial Derivatives
5.4. The Mean Value Theorem
5.5. Higher-Order Partial Derivatives and Taylor’s Theorem
5.6. Hypersurfaces and Tangent Hyperplanes in Rn
5.7. Max-Min Problems
5.8. Lagrange Multipliers
5.9. The Implicit and Inverse Function Theorems
5.10. Independent Projects
Chapter 6. Integration
6.1. Measures
6.2. Lebesgue Measure
6.3. Measurable Functions
6.4. The Integral
6.5. Lp Spaces
6.6. Fubini’s Theorem
6.7. Change of Variables in Integration
6.8. Independent Projects
Chapter 7. Fourier Analysis on Locally Compact Abelian Groups
7.1. Fourier Analysis on the Circle
7.2. Fourier Analysis on Locally Compact Abelian Groups
7.3. The Determination of ^G
7.4. The Fourier Transform on (R, +)
7.5. Fourier Inversion on (R, +)
7.6. Fourier Analysis on p-adic Fields
7.7. Independent Projects
Appendix A. Sets, Functions, and Other Basic Ideas
A.1. Sets and Elements
A.2. Equality, Inclusion, and Notation
A.3. The Algebra of Sets
A.4. Cartesian Products, Counting, and Power Sets
A.5. Some Sets of Numbers
A.6. Equivalence Relations and the Construction of Q
A.7. Functions
A.8. Countability and Other Basic Ideas
A.9. The Axiom of Choice
A.10. Independent Projects
Appendix B. Linear Algebra
B.1. Fundamentals of Linear Algebra
B.2. Linear Transformations
B.3. Linear Transformations and Matrices
B.4. Determinants
B.5. Geometric Linear Algebra
B.6. Independent Projects
Bibliography
Index of Terminology
Index of Notation Definitions
