分析（第1卷）
2012-9
世界图书出版公司
阿莫恩
426
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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：（德国）阿莫恩 （Herbert Amann）
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
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.
《分析(第1卷)(英文)》由世界图书出版公司北京公司出版。
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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。