实分析演讲集(英文)

实分析常以基础集合论、函数概念的定义等开始，专门研究数列、数列极限、微分、积分和函数序列，以及实函数的连续性、光滑性及其他相关性质。为了避免人们对实数先入为主的概念，本书把实数看作是一个完整有序域的元素。全书共十章，介绍了集合和函数，实数，开、闭和紧集，连续性，函数序列与函数系列，度量空间和收缩原理等内容，最后几章还对实分析的应用做了论述。为了学习者能进一步理解实分析的相关理论，作者还给出了多个不同难度的练习供读者参考。

Pretace
To the student
Chapter 1 Numbers,sets,and functions
1.1 The natural numbers,integers,and rational numbers
1.2 Sets
1.3 FLlnctions
More exercises
Chapter 2 The real numbers
2.1 The complete ordered field of real numbers
2.2 Consequences of completeness
2.3 Countable and uncountable sets
More exercises
Chapter 3 Sequences
3.1 Convergent sequences
3.2 New 1imits from old
3.3 Monotone sequences
3.4 Series
3.5 Subsequences and Cauchy sequences
More exercises
Chapter 4 Open,closed,and compact sets
4.1 Open and closed sets
4.2 Compact sets
More exercises
Chapter 5 Continuity
5.1 Limits of functions
5.2 Continuous functions
5.3 Continuous functions on compact sets and intervals
5.4 Monotone functions
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Chapter 6 Differentiation
6.1 Differentiable functions
6.2 The mean value theorem
More exercises
Chapter 7 Integration
7.1 The Riemann integral
7.2 The fundamental theorem of calculus
7.3 The natural logarithm and the exponential function
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Chapter 8 Sequences and series of functions
8.1 Pointwise and uniform convergence
8.2 Power series
8.3 Taylor series
8.4 The trigonometric functions
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Chapter 9 Metric spaces
9.1 Examples of metric spaces
9.2 Convergence and completeness in metric spaces
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Chapter 10 The contraction principle
10.1 The contraction principle
10.2 Picard's theorem
More exercises
Index
编辑手记