### 基本信息

- 原书名：Real Analysis with Economic Applications
- 原出版社： Princeton University Press

### 编辑推荐

是一部理想的教程和参考资料，填补了众多实分析教程不能帮助学生学习经济理论，帮助研究生接近经济学。

### 内容简介

目次：（第一部分）集合理论：实分析基础；可数性；（第二部分）矩阵空间上的分析：矩阵空间；连续性1；连续性2；（第三部分）线性空间上的分析：线性空间；凸性；经济中的应用；（第四部分）矩阵上的分析/赋范线性空间：矩阵线性空间；赋范线性空间；微积分。

读者对象：数学、经济学专业的高年级本科生、研究生和相关的科技人员。

### 目录

Preface xvii

Prerequisites xxvii

Basic Conventions xxix

PART I SET THEORY 1

CHAPTER A Preliminaries of Real Analysis

A.1 Elements of SetTheory 4

A.2 Real Numbers 33

A.3 Real Sequences 46

A.4 Real Functions 62

CHAPTER B Countability 82

B.1 Countable and Uncountable Sets 82

B.2 LosetsandQ 90

B.3 Some More Advanced Set Theory 93

B.4 Application: Ordinal Utility Theory 99

PART II ANALYSIS ON METRIC SPACES 115

CHAPTER C Metric Spaces 117

C.1 Basic Notions 118

C.2 Connectedness and Separability 138

C.3 Compactness 147

### 前言

My treatment is rigorous yet selective. I prove a good number of results here, so the reader will have plenty of opportunity to sharpen his or her understanding of the "theorem-proof" duality and to work through a variety of"deep" theorems of mathematical analysis. However, I take many short-cuts. For instance, I avoid complex numbers at all cost, assume compactness of things when one could get away with separability, introduce topological and topological linear concepts only via metrics or norms, and so on. My objective is not to report even the main theorems in their most general form

but rather to give a good idea to the student why these are true, or, even more important, why one should suspect that they must be true even before they are proved. But the shortcuts are not overly extensive in the sense that the main results covered here possess a good degree of applicability, especially for mainstream economics. Indeed, the purely mathematical development of the text is put to good use through several applications that provide con-cise introductions to a variety of topics from economic theory. Among these topics are individual decision theory, cooperative and noncooperative game theory, welfare economics, information theory, general equilibrium and finance, and intertemporal economics.

An obvious dimension that differentiates this text from various books on real analysis pertains to the choice of topics. I place much more empha-sis on topics that are immediately relevant for economic theory and omit some standard themes of real analysis that are of secondary importance for economists. In particular, unlike most treatments of mathematical analysis found in the literature, I present quite a bit on order theory, convex analysis,optimization, linear and nonlinear correspondences, dynamic program-ming, and calculus of variations. Moreover, apart from direct applications to economic theory, the exposition includes quite a few fixed point theorems,along with a leisurely introduction to differential calculus in Banach spaces.(Indeed, the second halfof the book can be thought of as providing a mod-est introduction to geometric (non)linear analysis.) However, because they play only a minor role in modern economic theory, I do not discuss topics such as Fourier analysis, Hilbert spaces, and spectral theory in this book.

While I assume here that the student is familiar with the notion of proof-this goal must be achieved during the first semester of a graduate economics program-I also spend quite a bit of time telling the reader why things are proved the way they are, especially in the earlier part of each chapter. At various points there are visible attempts to help the reader "see" a theo-rem (either by discussing informally the plan of attack or by providing a false-proof), in addition to confirming its validity by means of a formal proof. Moreover, whenever possible I have tried to avoid rabbit-out-of-the-hat proofs and rather give rigorous arguments that explain the situation that is being analyzed. Longer proofs are thus often accompanied by foot-notes that describe the basic ideas in more heuristic terms, reminiscent of how one would "teach" the proof in the classroom.1 This way the mate-rial is hopefully presented at a level that is readable for most second- or third-semester graduate students in economics and advanced undergrad-uates in mathematics while still preserving the aura of a serious analysis course. Having said this, however, I should note that the exposition gets less restrained toward the end of each chapter, and the analysis is presented without being overly pedantic. This goes especially for the starred sections,which cover more advanced material than the rest of the text.

The basic approach is, of course, primarily that of a textbook rather than a reference. But the reader will still find here the careful yet unproved statements of a good number of "difficult" theorems that fit well with the overall development; some examples include Blumberg's Theorem, non- contractibility of the sphere, Rademacher's Theorem on the differentiability of Lipschitz continuous functions, Motzkin's Theorem, and Reny's Theo- rem on the existence of the Nash equilibrium. At the very least, this should hint to the student what might be expected in a higher-level course. Fur-thermore, some of these results are widely used in economic theory, so it is desirable that the students begin at this stage developing a precursory under-standing of them. To this end, I discuss some of these results at length, talk about their applications, and at times give proofs for special cases. It is worth noting that the general exposition relies on a select few of these results.

Last but not least, it is my sincere hope that the present treatment pro-vides glimpses of the strength of abstract reasoning, whether it comes from applied mathematical analysis or from pure analysis. I have tried hard to strike a balance in this regard. Overall, I put far more emphasis on the appli-cability of the main theorems relative to their generalizations or strongest formulations, only rarely mention if something can be achieved without invoking the Axiom of Choice, and use the method of proof by contradic-tion more frequently than a "purist" might like to see. On the other hand,by means of various remarks, exercises, and the starred sections, I touch on a few topics that carry more of a pure mathematician's emphasis. (Some examples here include the characterization of metric spaces with the Banach fixed point property, the converse of Weierstrass' Theorem, various charac-terizations of infinite-dimensional normed linear spaces, and so on.) This reflects my full agreement with the following wise words of Tom K6rner:

A good mathematician can look at a problem in more than one way.In particular, a good mathematician will "think like a pure mathemati-cian when doing pure mathematics and like an applied mathematician when doing applied mathematics." (Great mathematicians think like themselves when doing mathematics.)2

On the Structure of the Text

This book consists of four parts:

I. Set Theory (Chapters A and B)

II. Analysis on Metric Spaces (Chapters C, D, and E)

III. Analysis on Linear Spaces (Chapters F, G, and H)

IV. Analysis on Metric/Normed Linear Spaces (Chapters I, J, and K)

Part I provides an elementary yet fairly comprehensive overview of (intuitive) set theory. Covering the fundamental notions of sets, relations,functions, real sequences, basic calculus, and countability, this part is a prerequisite for the rest of the text. It also introduces the Axiom of Choice and some of its equivalent formulations, and sketches a brief introduction to order theory. Among the most notable theorems covered here are Tarski's Fixed Point Theorem and Szpilrajn's Extension Theorem.

Part II is (almost) a standard course on real analysis on metric spaces.It studies at length the topological properties of separability and compact-ness and the uniform property of completeness, along with the theory ofcontinuous functions and correspondences, in the context of metric spaces.I also talk about the elements of fixed point theory (in Euclidean spaces) and introduce the theories of stationary dynamic programming and Nash equi-librium. Among the most notable theorems covered here are the Contraction Mapping Principle, the Stone-Weierstrass Theorem, the Tietze Extension Theorem, Berge's Maximum Theorem, the fixed point theorems of Brouwer and Kakutani, and Michael's Selection Theorem.

Part III begins with an extensive review of some linear algebraic con-cepts (such as linear spaces, bases and dimension, and linear operators),then proceeds to convex analysis. A purely linear algebraic treatment of both the analytic and geometric forms of the Hahn-Banach Theorem is given here, along with several economic applications that range from individual decision theory to financial economics. Among the most notable theo-rems cove'red are the Hahn-Banach Extension Theorem, the Krein-Rutman Theorem, and the Dieudonne Separation Theorem.

Part IV can be considered a primer on geometric linear and nonlinear analysis. Since I wish to avoid the consideration of general topology in this text, the entire discussion is couched within metric and/or normed linear spaces. The results on the extension of linear functionals and the separation by hyperplanes are sharpened in this context, an introduction to infinite-dimensional convex analysis is outlined, and the fixed point the- ory developed earlier within Euclidean spaces is carried into the realm of normed linear spaces. The final chapter considers differential calculus and optimization in Banach spaces and, by way of an application, provides an introductory but rigorous approach to calculus of variations. Among the most notable theorems covered here are the Separating Hyperplane Theorem, the Uniform Boundedness Principle, the Glicksberg-Fan Fixed Point Theorem, Schauder's Fixed Point Theorems, and the Krein-Milman Theorem.

On the Exercises

As in most mathematics textbooks, the exercises provided throughout the text are integral to the ongoing exposition and hence appear within the main body of various sections. Many of them appear after the introduction of a particularly important concept to make the reader better acquainted with that concept. Others are given after a major theorem in order to illustrate how to apply the associated result or the method of proof that yielded it.