实分析及其在经济学中的应用

Preface

Prerequisites

Basic Conventions

PART Ⅰ SET THEORY

CHAPTER A Preliminaries of Real Analysis

A.1 Elements ofSet Theory

A.1.1 Sets

A.1.2 Relations

A.1.3 Equivalence Relations

A.1.4 O0rder Relations

A.1.5 Functions

A.1.6 Sequences, Vectors, and Matrices

A.1.7 A Glimpse ofAdvanced Set Theory: The Axiom of Choice

A.2 Real Numbers

A.2.1 Ordered Fields

A.2.2 Natural Numbers, Integers, and Rationals

A.2.3 Real Numbers

A.2.4 Intervals and R

A.3 Real Sequences

A.3.1 Convergent Sequences

A.3.2 Monotonic Sequences

A.3.3 Subsequential Limits

A.3.4 Infinite Series

A.3.5 Rear.rangement oflnfinite Series

A.3.6 Infinite Products

A.4 Real Functions

A.4.1 Basic Definitions

A.4.2 Limits, ContinLuty, and Differentiation

A.4.3 Riemann Integration

A.4.4 Exponential, Logarithmic, and Trigonometric Functions

A.4.5 Concave and Convex Functions

A.4.6 Quasiconcave and Quasiconvex Functions

CHAPTER B Countability

B.1 Countable and Uncountable Sets

B.2 Losets and Q

B.3 Some More Advanced Set Theory

B.3.1 The Cardinality Ordering

B.3.2 The Well-Ordering Principle

B.4 Application: Ordinal utility Theor）r

B.4.1 Preference Relations

B.4.2 Utilitv ReDresentation of Complete Preference Relations

B.4.3 Utility Representation oflncomplete Preference Relations

PART Ⅱ ANALYSIS ON METRIC SPACES

CHAPTER C Metric Spaces

C.1 Basic Notions

C.1.1 Metric Spaces: Definition and Examples

C.1.2 0pen and Closed Sets

C.1.3 Convergent Sequences

……

PART Ⅲ ANALYSIS ON LINEAR SPACES

PART Ⅳ ANALYSIS ON METRIC/NORMED LINEAR SPACES

Hints for Selected Exercises

References

Clossary of Selected Symbols

Index

This is primarily a textbook on mathematical analysis for graduate students in economics. While there are a large number of excellent textbooks on this broad topic in the mathematics literature， most ofthese texts are overly advanced relative to the needs of the vast majority of economics students and concentrate on various topics that are not readily helpful for studying economic theory. Moreover， it seems that most economics students lack the time or courage to enroll in a math course at the graduate level. Sometimes this is not even for bad reasons， for only few math departments offer classes that are designed for the parhcular needs of economists. Unfortunately，more often than not， the consequent lack ofmathematical background cre-ates problems for the students at a later stage of their education， since an exceedingly large fraction ofeconomic theory is impenetrable without some rigorous background in real analysis. The present text aims at providing a remedy for this inconvenient situation.