分析（第1卷）

內(nèi)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.

作者簡介

作者：（德國）阿莫恩 （Herbert Amann）

書籍目錄

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

章節(jié)摘錄

版權(quán)頁：   插圖：   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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