应用数学分析外文翻译资料

应用数学分析

原文作者：John K.Hunter,Brnuo Nachtergaele

单位：加州大学戴维斯分校

摘要：数学知识和复杂性、计算能力和应用领域是一个巨大的速度扩张。因此,应用数学家的培训的要求也在提高。因此不容易决定什么应该构成核心应用数学家的数学训练。我们认为每一个应用数学家,无论他或她的终极领域感兴趣的可能是,应该有一个接地的基本面分析。这本书,材料已超出所能覆盖在一年开始研究生课程。教学三个季度课程时,我们通常在第一季度,封面1-5章提供的先进的计算和讨论度量和赋范空间的基本性质,其次是6-9章在第二季度,它专注于希尔伯特空间,包括傅里叶级数和有界的线性运算符。在最后一节中,我们将介绍选择的主题从10-13章,讨论绿色的功能,无限运营商分布理论,傅里叶变换、测量理论、函数空间,微分学在巴拿赫空间中。主题的选择和重点取决于学生的背景和兴趣。

关键词：度量； 微积分；分布理论

前言

数学知识以及数学的复杂性、数学的计算能力、应用领域等，正在以极快的速度扩展。因此，对应用数学家的培养需求也一直在增长。但是，要说对一个应用数学家在数学上的核心培养内容应该是什么，真的很不容易。我们认为，对每一个应用数学家来说，无论他/她最终感兴趣的领域是什么，他/她都应该掌握基础的分析知识。

本书的目的在于，为低年级研究生介绍在应用领域最有用的几部分分析知识。我们的素材都是根据它在应用问题上的使用而选的，并尽我们所能以简单明了的方式呈现出来，当然，我们也不会失去该有的数学上的严谨。

我们注重基本概念的介绍，尽量避免专业性更强的领域的一些术语。虽然我们在尽一切努力促进我们所介绍的理论的发展，也在书中包含了不同领域的各种各样的例子，但是这本书最重要的还是关于分析。

我们不要求读者有广泛的数学预备知识。本书旨在能被众多不同学科背景的学生所接受，包括从非数学专业转入数学专业的本科毕业生，以及科学和工程专业那些想要学习数学分析的研究生。只要你懂基本的微积分，线性代数，常微分方程以及对函数和设置有一些了解就足够了。我们偶尔也会使用一些复杂分析的初步结果，但我们不会在文中去展开任何复杂分析的方法过程。

我们为主要的论题提供了详细的论据。我们没有尝试从最大的普遍性来解释结果，而是以简单、具体的方式来阐述我们的主要观点。我们经常从不同的形式背景得到相同的观点，即使这导致一些对先前的定义和结果的重复。我们广泛使用例子和习题来阐明我们介绍的理论、观点。这些习题难易程度不等。有些是基础题，尽管我们已经省略了很多在上课时布置过的常规练习。还有些比较难一点的，用来介绍我们在主文中没有提到的新概念或者新应用。

有一块我们没有完全说明的知识是Lebesgue测度和积分。因为，测量理论的全面发展还需要我们付出很多努力，而且，不管怎样，Lebesgue积分的使用比建立更容易。

在写这本书的过程中，我们使用的材料已经超出低年级研究生一年的课程所覆盖的内容。在教授三个季度的课程时，我们通常在第一季度教授1-5章，这五章内容复习了高级微积分，讨论了度量和赋范空间的基本性质。在第二季度，我们会教授6-9章，这几章主要讲述希尔伯特空间，包括傅里叶级数和有界的线性运算符。在最后一个季度,我们会从10-13章中选出一些题目来讲,包括格林函数,无界算子,分布理论，傅里叶变换，测量理论，函数空间,以及巴拿赫空间微分学。主题的选择和重点取决于学生的背景和兴趣。

书中所用的材料都是权威的。我们引用的许多来源已在参考文献中列出。但是，参考文献并不全面，仅限于我们觉得对本书的受众有用的那些文章，包括作为背景知识的阅读，或者为了更深程度地了解相关主题。

感谢选择了这门课并对课程内容初稿提供了宝贵建议的学生们。另外，特别感谢Scott Beaver，Sergio Lucero以及John Thoo在数据准备和手稿校验上给予的莫大帮助！

第一章

度量和赋范空间

我们都熟悉普通的几何性质,三维欧几里得空间。一个永恒的主题在数学是各种对象的分组成抽象的空间。这个分组使我们扩展我们的直觉在欧几里得空间点之间的关系更一般的类对象之间的关系,从而更清晰和深入了解的对象。

最简单的设置许多问题分析研究是一个度量空间。度量空间是一个点集与一个合适的点之间的距离的概念。我们可以使用公制,或距离函数,定义分析的基本音乐会,如收敛性、连续性和密实度。

一种度量空间不需要任何代数结构定义。然而,在许多应用程序中,度量空间是一个线性空间与来自标准的度量,给出了向量的“长度”。这样的空间被称为赋范线性空间。例如,n维欧氏空间是一个赋范线性空间(在任意点为原点)的选择。这本书的中心主题是华氏赋范线性空间的研究,包括功能空间的一个点表示一个函数。正如我们将看到的,几何直觉来源于有限维欧氏空间仍然是至关重要的,尽管完全新的特性出现在华氏空间。

在这一章中,我们定义酸研究度量空间和赋范线性空间。一路上,我们回顾一些定义酸实分析的结果。

第二章

连续函数

本章中,我们介绍了赋范线性空间的概念,与有限维欧氏空间R”为主要例子。在这一章,我们研究线性空间上的连续函数紧集配备了统一的规范。这些功能空间是我们的第一个例子的无限维赋范线性空间,我们探索收敛的概念,完整性、密度、和密实度。作为一个紧凑的应用,我们证明存在结果为常微分方程初值问题。

第三章

收缩映射定理

在这一章里,我们国家和收缩映射定理证明,这是一种最简单和最有用的方法为线性和非线性方程组的建设解决方案。我们也存在一些定理的应用。

第四章

拓扑空间

在前面的章节中,我们讨论了序列的收敛性,函数的连续性,集的密实度。我们对这些属性的指标或标准。某些类型的融合,如实值函数的收敛点上定义一个间隔,不能表达了一个函数的度量空间。拓扑空间提供了一个总体框架研究收敛性,连续性,和紧凑性。拓扑空间的基本结构并不是一个距离函数,但开集的集合;直接开集的角度思考往往导致更清晰和更大的通用性。

第五章

巴拿赫空间

许多线性方程组可能制定的一个合适的巴拿赫空间上线性算子的表演。在这一章,我们研究巴拿赫空间和线性oper-ators巴拿赫空间更详细地行事。我们给巴拿赫空间的定义,并说明的例子。我们表明,线性算子是连续当且仅当它是有界的,定义一个有界的线性算子的规范,研究线性算子的一些性质有界。无界线性运营商应用也很重要:例如,微分运算符通常是无限的。在后面的章节中,我们将研究他们在简单的希尔伯特空间的上下文。

外文文献出处：John K.Hunter,Brnuo Nachtergaele《Applied Analysis》

附外文文献原文

Applied Analysis

John K.Hunter,Brnuo Nachtergaele

Department of Mathematics University of California at Davis

September 12,2000

Preface

Mathematical knowledge and sophistication, computational power, and areas of application are expanding at an enormous rate. As a result, the demands on the training of applied mathematicians are increasing all the time. It is therefore not easy to decide what should constitute the core mathematical training of an applied mathematician. We take the view that every applied mathematician, whatever his or her ultimate area of interest may turn out to be, should have a grounding in the fundamentals of analysis.

The aim of this book is to supply an introduction for beginning graduate students to those parts of analysis that are most useful in applications. The material is selected for its use in applied problems, and is presented as clearly and simply as we are able, but without the sacrifice of mathematical rigor.

We focus on ideas of central importance, and attempt to avoid technicalities and detours into areas of more specialized interest. While we make every effort to motivate the areas introduced, and include a variety of examples from different fields, this book is first and foremost about analysis. We do not assume extensive mathematical prerequisites of the reader. The book is intended to be accessible to students from a wide variety of backgrounds, including undergraduate students entering applied mathematics from non-mathematical fields, and graduate students in the sciences and engineering who would like to learn analysis. A basic background in calculus, linear algebra, ordinary differential equations, and some familiarity with functions and sets should be sufficient. We occasionally use some elementary results from complex analysis, but we do not develop any methods from complex analysis in the text.

We provide detailed proofs for the main topics. We make no attempt to state results is maximum generality, and instead illustrate the main ideas in simple, concrete settings. We often return to the same ideas in different contexts, even if this leads to some repetition of previous definitions and results. We make extensive use of examples and exercises to illustrate the concepts introduced. The exercises are at various levels; some are elementary, although we have omitted many of the routine exercises that we assign while teaching the class, and some are harder and are an excuse to introduce new ideas or applications not covered in the main text. One area where we do not give a complete treatment is Lebesgue measure and integration. A full development of measure theory would take us too far afield, and, in any event, the Lebesgue integral is much easier to use than to construct.

In writing this book, the material has expanded beyond what can be covered in a year

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Applied Analysis

John K.Hunter,Brnuo Nachtergaele

Department of Mathematics University of California at Davis

September 12,2000

Preface

Mathematical knowledge and sophistication, computational power, and areas of application are expanding at an enormous rate. As a result, the demands on the training of applied mathematicians are increasing all the time. It is therefore not easy to decide what should constitute the core mathematical training of an applied mathematician. We take the view that every applied mathematician, whatever his or her ultimate area of interest may turn out to be, should have a grounding in the fundamentals of analysis.

The aim of this book is to supply an introduction for beginning graduate students to those parts of analysis that are most useful in applications. The material is selected for its use in applied problems, and is presented as clearly and simply as we are able, but without the sacrifice of mathematical rigor.

We focus on ideas of central importance, and attempt to avoid technicalities and detours into areas of more specialized interest. While we make every effort to motivate the areas introduced, and include a variety of examples from different fields, this book is first and foremost about analysis. We do not assume extensive mathematical prerequisites of the reader. The book is intended to be accessible to students from a wide variety of backgrounds, including undergraduate students entering applied mathematics from non-mathematical fields, and graduate students in the sciences and engineering who would like to learn analysis. A basic background in calculus, linear algebra, ordinary differential equations, and some familiarity with functions and sets should be sufficient. We occasionally use some elementary results from complex analysis, but we do not develop any methods from complex analysis in the text.

We provide detailed proofs for the main topics. We make no attempt to state results is maximum generality, and instead illustrate the main ideas in simple, concrete settings. We often return to the same ideas in different contexts, even if this leads to some repetition of previous definitions and results. We make extensive use of examples and exercises to illustrate the concepts introduced. The exercises are at various levels; some are elementary, although we have omitted many of the routine exercises that we assign while teaching the class, and some are harder and are an excuse to introduce new ideas or applications not covered in the main text. One area where we do not give a complete treatment is Lebesgue measure and integration. A full development of measure theory would take us too far afield, and, in any event, the Lebesgue integral is much easier to use than to construct.

In writing this book, the material has expanded beyond what can be covered in a year long course for beginning graduate students. When teaching a three quarter course, we usually cover Chapters 1-5 in the first quarter, which provide a review of advanced calculus and discuss the basic properties of metric and normed spaces, followed by Chapters 6-9 in the second quarter, which focus on Hilbert spaces, including Fourier series and bounded linear operators. In the last quarter, we cover a selection of topics from Chapters 10-13, which discuss Greens functions, unbounded operators, distribution theory, the Fourier transform, measure theory, function spaces, and differential calculus in Banach spaces. The choice and emphasis of the topics depends on the backgrounds and interests of the students.

The material presented here is standard. Many of the sources we have drawn upon are listed in the bibliography. The bibliography is not comprehensive, however, and is limited to books that we feel will be useful to the intended audience of this text, either for background reading, or to pursue in greater depth some of the topics treated here.

We thank the students who have taken this course and contributed comments and suggestions on early drafts of the course notes. In particular, Scott Beaver, Sergio Lucero, acid John Thoo were helpful in the preparation of figures and the proofreading of the manuscript.

Chapter 1

Metric and Normed Spaces

We are all familiar with the geometrical properties of ordinary, three dimensional Euclidean space. A persistent theme in mathematics is the grouping of various kinds of objects into abstract spaces. This grouping enables us to extend our intuition of the relationship between points in Euclidean space to the relationship between more general kinds of objects, leading to a clearer and deeper understanding of those objects.

The simplest setting for the study of many problems in analysis is that of a metric space. A metric space is a set of points with a suitable notion of the distance between points. We can use the metric, or distance function, to define the fundamental concerts of analysis, such as convergence, continuity, and compactness.

A metric space need not have any kind of algebraic structure defined on it. In many applications, however, the metric space is a linear space with a metric derived from a norm that gives the 'length' of a vector. Such spaces are called normed linear spaces. For example, n-dimensional Euclidean space is a normed linear space (after the choice of an arbitrary point as the origin). A central topic of this book is the study of infinite-dimensional normed linear spaces, including function spaces in which a single point represents a function. As we will see, the geometrical intuition derived from finite-dimensional Euclidean space remains essential, although completely new features arise in the case of infinite-dimensional spaces.

In this chapter, we define acid study metric spaces and normed linear spaces. Along the way, we review

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