本书是当代数学大师的著作。该书重点论述一致收敛、一到极限，以及在积分或微分情况下普遍的一致性等理论。

这本教材的蓝本从1968年开始使用，先后两次改版，重印四次，非常适合学过微积分的高校数学系本科生使用。

本书作者是当代数学大师，这本教材的蓝本从1968年开始使用，先后两次改版，重印四次，非常适合学过微积分的高校数学系本科生使用。本书重点论述一致收敛、一到极限，以及在积分或微分情况下普遍的一致性等理论。

Foreword to the First Edition

Foreword to the Second Edition

PART ONE

Review of Calculus

CHAPTER 0

Sets and Mappings

1. Sets

2. Mappings

3. Natural Numbers and Induction

4. Denumerable Sets

5. Equivalence Relations

CHAPTER I

Real Numbers

1. Algebraic Axioms

2. Ordering Axioms

3. Integers and Rational Numbers

4. The Completeness Axiom

CHAPTER II

Limits and Continuous Functions

1. Sequences of Numbers

2. Functions and Limits

3. Limits with Infinity

4. Continuous Functions

CHAPTER III

Differentiation

1. Properties of the Derivative

2. Mean Value Theorem

3. Inverse Functions

CHAPTER IV

Elementary Functions 

I. Exponential

2. Logarithm

3. Sine and Cosine

4. Complex Numbers

CHAPTER V

The Elementary Real Integral

1. Characterization of the Integral

2. Properties of the Integral

3. Taylor's Formula

4. Asymptotic Estimates and Stirling's Formula

PART TWO

Convergence

CHAPTER VI

Normed Vector Spaces

1. Vector Spaces

2. Normcd Vector Spaces

3. n-Space and Function Spaces

4. Completeness

5. Open and Closed Sets

CHAPTER VII

Limits

1. Basic Properties

2. Continuous Maps

3. Limits in Function Spaces

4. Completion of a Normed Vector Space

CHAPTER VIII

Compactness 

L Basic Properties of Compact Sets

2. Continuous Maps on Compact Sets

3. Algebraic Closure of the Complex Numbers

4. Relation with Open Coverings

CHAPTER IX

Series

1. Basic Definitions 

2. Series of Positive Numbers

3. Non-Absolute Convergence

4. Absolute Convergence in Vector Spaces

5. Absolute and Uniform Convergence

6. Power Series

7. Differentiation and Integration of Series

CHAPTER X

The Integral In One Variable 

1. Extension Theorem for Linear Maps

2. Integral of Step Maps

3. Approximation by Step Maps

4. Properties of the IntegralAppendix. The Lebesgue Integral 

5. The Derivative

6. Relation Between the Integral and the Derivative

7. Interchanging Derivatives and Integrals

PART THREE

Applications of the Integral

CHAPTER XI

Approximation with Convolutions

1. Dirac Sequences

2. The Weierstrass Theorem

CHAPTER XII

Fourier Series

1. Hermitian Products and Orthogonality

2. Trigonometric Polynomials as a Total Family

3. Explicit Uniform Approximation

4. Pointwise Convergence

CHAPTER XIII

Improper Integrals

1. Definition

2. Criteria for Convergence

3. Interchanging Derivatives and Integrals

4. The Heat Kernel

CHAPTER XIV

The Fourier Integral

1. The Schwartz Space

2. The Fourier Inversion Formula

3. An Example of Fourier Transform not in the Schwartz Space

PART FOUR

Calculus In Vector Spaces

CHAPTER XV

Functlona on n-Space

1. Partial Derivatives

2. Differentiability and the Chain Rule

3. Potential Functions

4. Curve Integrals

5. Taylor's Formula

6. Maxima and the Derivative

CHAPTER XVI

The Winding Number and Global Potential Functions

I. Another Description of the Integral Along a Path

2. The Winding Number and Homology

3. Proof of the Global Integrability Theorem

4. The Integral Over Continuous Paths

5. The Homotopy Form of the Integrability Theorem

6. More on Homotopies

CHAPTER XVII

Derivatives In Vector Spaces

1. The Space of Continuous Linear Maps

2. The Derivative as a Linear Map

3. Properties of the Derivative

4. Mean Value Theorem

5. The Second Derivative

6. Higher Derivatives and Taylor's Formula

7. Partial Derivatives

8. Differentiating Under the Integral Sign

CHAPTER XVIII

Inverse Mapping Theorem

I. The Shrinking Lemma

2. Inverse Mappings, Linear Case

3. The Inverse Mapping Theorem

4. Implicit Functions and Charts

5. Product Decompositions

CHAPTER XIX

Ordinary Differential Equations

1. Local Existence and Uniqueness

2. Approximate Solutions

3. Linear Differential Equations

4. Dependence on Initial Conditions

PART FIVE

Multiple Integration

CHAPTER XX

Multiple Integrals

1. Elementary Multiple Integration

2. Criteria for Admissibility

3. Repeated Integrals

4. Change of Variables

5. Vector Fields on Spheres

CHAPTER XXI

Differential Forms

1. Definitions

2. Stokes' Theorem for a Rectangle

3. Inverse Image of a Form

4. Stokes" Formula for Simplices

Appendix

Index