本书是自世界著名大学剑桥大学出版社引进的英文版数学教程,中文书名可译为《实分析演讲集》作者是芬纳.拉尔森教授,他任教于澳大利亚阿德莱德大学。剑桥大学出版社对本书的介绍是这样写的:
本书是为本科学生准备的对实分析的严谨的介绍,从全序域的定理和一些集合论知识开始。本书避免了任何关于实数的先入之见,只把它们当作全序域的元素来研究。包括所有的标准主题,以及对三角函数的适当处理,许多人认为这些内容都是理所应当的。书的最后几章提供了一个详细的、基于实例的对应用在实线上的微分方程的度量空间的介绍。
作者的阐述简明扼要,帮助学生抓住要点。本书包括200多个不同难度的练习题,其中许多题都涉及正文中的理论内容。该书非常适合本科二年级学生和需要掌握实分析基础知识的更高年级的学生阅读。
This book is a rigorous introduction to real analysis, suitable for a one semester course at the second-year undergraduate level, based on my experience of teaching this material many times in Australia and Canada. My aim is to give a treatment that is brisk and concise, but also reasonably complete and as rigorous as is practicable, starting from the axioms for a complete ordered field and a little set theory.
Along with epsilons and deltas, I emphasise the alternative language of neighbourhoods, which is geometric and intuitive and provides an introduction to topological ideas. I have included a proper treatment of the trigonometric functions. They are sophisticated objects, not to be taken for granted. This topic is an instructive application of the theory of power series and other earlier parts of the book. Also, it involves the concept of a group, which most students won't have seen in the context of analysis before.
目录(翻译如下)
1.数,集合与函数(自然数,整数和有理数,集合,函数)
2.实数(实数的全序域,完整性的结果,可数集合与非可数集合)
3.序列(收敛数列,单调序列,级数,子序列和柯西序列)
4.开集,闭集和紧集
5.连续性(函数的极限,连续函数,紧集和区间上的连续函数,单调函数)
6.微分(微分函数,中值定理)
7.积分(微积分基本定理,黎曼积分,自然对数和指数函数)
8.函数的序列和级数(点态收敛与一致收敛,幂级数,泰勒级数,圆函数)
9.度量空间(度量空间的例子,度量空间的收敛与完整性)
10.收缩原理(热胀冷缩的原理,毕卡(Picard)定理)