分析基礎

《對於學習高等分析教程，復變量、微分方程、傅里葉分析、數值分析、多變量微積分以及統計的人來説讀《分析基礎:微積分理論》是很有必要的，也可以作為將來中學老師的參考教科書。有關實線的概念《分析基礎:微積分理論》研究不多，許多抽象概念，如矩陣空間和有序系，在《分析基礎:微積分理論》中都避免討論；最小上界的性質在《分析基礎:微積分理論》中拿來即用，實線的有序性貫穿於書的始終；數字序列的透徹講解作為學習標準微積分的基礎；選修部分可以讓學生學習矩陣空間和Riemann-Stieltjes積分。 ^[2] 

作者：(美國)丁羅斯(Ross.K.A)

Preface

1 Introduction

1 The Set N of Natural Numbers

2 The Set Q of Rational Numbers

3 The Set R of Real Numbers

4 The Completeness Axiom

5 The Symbols +oo and -oo

6 * A Development of R

2 Sequences

7 Limits of Sequences

8 A Discussion about Proofs

9 Limit Theorems for Sequences

10 Monotone Sequences and Cauchy Sequences

11 Subsequences

12 lira sup's and lim inf's

13 * Some Topological Concepts in Metric Spaces

14 Series

15 Alternating Series and Integral Tests

16 * Decimal Expansions of Peal Numbers

3 Continuity

17 Continuous Functions

18 Properties of Continuous Functions

19 Uniform Continuity

20 Limits of Functions

21 * More on Metric Spaces: Continuity

22 * More on Metric Spaces: Connectedness

4 Sequences and Series of Functions

23 Power Series

24 Uniform Convergence

25 More on Uniform Convergence

26 Differentiation and Integration of Power Series

27 * Weierstrass's Approximation Theorem

Differentiation

28 Basic Properties of the Derivative

29 The Mean Value Theorem

30 * UHospital's Rule

31 Taylor's Theorem

5 Integration

32 The Riemann Integral

33 Properties of the Riemann Integral

34 Fundamental Theorem of Calculus

35 * Riemann-Stieltjes Integrals

36 * Improper Integrals

37 * A Discussion of Exponents and Logarithms

Appendix on Set Notation

Selected Hints and Answers

References

8ymhola Index

Index