分析（第1卷）

出版时间：2012-9  出版社：世界图书出版公司  作者：阿莫恩  页数：426  
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内容概要

　　The present book strives for clarity and transparency. Right
from the begin-ning， it requires from the reader a willingness to
deal with abstract concepts， as well as a considerable measure of
self-initiative. For these e&，rts， the reader will be richly
rewarded in his or her mathematical thinking abilities， and will
possess the foundation needed for a deeper penetration into
mathematics and its applications.
　　This book is the first volume of a three volume introduction to
analysis. It de- veloped from. courses that the authors have taught
over the last twenty six years at the Universities of Bochum， Kiel，
Zurich， Basel and Kassel. Since we hope that this book will be used
also for self-study and supplementary reading， we have included far
more material than can be covered in a three semester sequence.
This allows us to provide a wide overview of the subject and to
present the many beautiful and important applications of the
theory. We also demonstrate that mathematics possesses， not only
elegance and inner beauty， but also provides efficient methods for
the solution of concrete problems.

作者简介

作者：（德国）阿莫恩 （Herbert Amann）

书籍目录

Preface
　Chapter Ⅰ Foundations
　1 Fundamentals of Logic
　2 Sets
　Elementary Facts
　The Power Set
　Complement, Intersection and Union
　Products
　Families of Sets
　3 Functions，
　Simple Examples
　Composition of Functions
　Commutative Diagrams
　Injections, Surjections and Bijections
　Inverse Functions
　Set Valued Functions
　4 Relations and Operations
　Equivalence Relations
　Order Relations
　Operations
　5 The Natural Numbers
　The Peano Axioms
　The Arithmetic of Natural Numbers
　The Division Algorithm
　The Induction Principle
　Recursive Definitions
　6 Countability
　Permutations
　Equinumerous Sets
　Countable Sets
　Infinite Products
　7 Groups and Homomorphisms
　Groups
　Subgroups
　Cosets
　Homomorphisms
　Isomorphisms
　8 R.ings, Fields and Polynomials
　Rings
　The Binomial Theorem
　The Multinomial Theorem
　Fields
　Ordered Fields
　Formal Power Series
　Polynomials
　Polynomial Functions
　Division of Polynomiajs
　Linear Factors
　Polynomials in Several Indeterminates
　9 The Rational Numbers
　The Integers
　The Rational Numbers
　Rational Zeros of Polynomials
　Square Roots
　10 The Real Numbers
　Order Completeness
　Dedekind's Construction of the Real Numbers
　The Natural Order on R
　The Extended Number Line
　A Characterization of Supremum and Infimum
　The Archimedean Property
　The Density of the Rational Numbers in R
　nth Roots
　The Density of the Irrational Numbers in R
　Intervals
Chapter Ⅱ Convergence
Chapter Ⅲ Continuous Functions
Chapter Ⅳ Differentiation in One Variable
Chapter Ⅴ Sequences of Functions
Appendix Introduction to Mathematical Logic
Bibliography
Index

章节摘录

版权页：   插图：   In this chapter, approximations are once again the center of our interest. Just as in Chapter Ⅱ, we study sequences and series. The difference is that we consider here the more complex situation of sequences whose terms are functions. In this circumstance there are two viewpoints: We can consider such sequences locally,that is, at each point, or globally. In the second case it is natural to consider the terms of the sequence as elements of a function space so that we are again in the situation of Chapter Ⅱ. If the functions in the sequence are all bounded, then we have a sequence in the Banach space of bounded functions, and we can apply all the results about sequences and series which we developed in the second chapter.This approach is particularly fruitful, allows short and elegant proofs, and, for the first time, demonstrates the advantages of the abstract framework in which we developed the fundamentals of analysis. In the first section we analyze the various concepts of convergence which appea in the study of sequences of functions. The most important of these is uniform convergence which is simply convergence in the space of bounded functions. The main result of this section is the Weierstrass majorant criterion which is nothing more than the majorant criterion from the second chapter applied to the Banach space of bounded functions. Section 2 is devoted to the connections between continuity, difFerentiability and convergence for sequences of functions. To our supply of concrete Banach spaces,we add one extremely important and natural example: the space of conthinuous functions on a compact metric space.

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《分析(第1卷)(英文)》由世界图书出版公司北京公司出版。

用户评论 (总计4条)

•   新的德國分析書, 內容廣泛!
•   德国数学家Herbert Amann的经典数学分析专著与众不同，和俄罗斯数学家菲赫金哥尔茨的“数学分析原理”，“微积分教程”等传统数学分析专著完全不同，是从集合论入手讲述数学分析，逻辑性特别强，非常严密，适合数学专业的大学生阅读,经典.
•   我是调查了之后，才决定购买的。据说这是观点最新数学分析的教材，在欧洲最受推崇
•   这套书给人的感觉有点不上不下。具体来说，作者（基本上是）打算避开集合论公理和数理逻辑，但又花了十几页的功夫去描述这两个东西，而且还是在避免使用符号语言的情况下，使用自然语言来说明的.......嘛，因为原文是德文，说明上应该会比这英译本的要严格一些，但是这英译本就......举个例子来讲，英译本中一会儿用英语“and”来表示逻辑符号里的"AND"，一会儿又用“and”来表示逻辑符号里的"INCLUSIVE OR"。都无语了......书中有些证明需要构造算子或代数结构，但由于书中没有给出wff的相关逻辑规则，实际上，这些算子和代数结构不应该要求读者去构造。因为这本书并没有告诉读者如何去检验自己的构造是否合理。