內容簡介
分析基礎《分析基礎:微積分理論》旨在為對嚴格的證明不熟悉的讀者，提供一本學習標準積分教程的後繼教程。對於學習高等分析教程，復變數、微分方程、傅立葉分析、數值分析、多變數微積分以及統計的人來說讀《分析基礎:微積分理論》是很有必要的。同樣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。
目錄
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 Completenessaxiom
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
12lirasup's and lim inf's
13 * Some Topological Concepts in Metric Spaces
14 Series
15 Alternating Series and Integral Tests
16 * Decimal Expansions ofpealNumbers
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
