Real Analysis: A Constructive Approach (Pure and Applied Mathematics: A Wiley-Interscience Series of实分析：建设性研究

作者： Mark Bridger 著

出 版 社： 吉林长白山

出版时间： 2006-11-1字数：版次： 1页数： 302印刷时间： 2006/11/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780471792307包装： 精装内容简介

This innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. The ultimate consequence of this method is that it makes sense—whether you’re a math major or student in any branch of the sciences.

This innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense—not just to math majors but also to students from all branches of the sciences.

The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes:

Early use of the Completeness Theorem to prove a helpful Inverse Function Theorem

Sequences, limits and series, and the careful derivation of formulas and estimates for important functions

Emphasis on uniform continuity and its consequences, such as boundedness and the extension of uniformly continuous functions from dense subsets

Construction of the Riemann integral for functions uniformly continuous on an interval, and its extension to improper integrals

Differentiation, emphasizing the derivative as a function rather than a pointwise limit

Properties of sequences and series of continuous and differentiable functions

Fourier series and an introduction to more advanced ideas in functional analysis

Examples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging.

This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences.

作者简介：

MARK BRIDGER, PHD, is Associate Professor of Mathematics at Northeastern University in Boston, Massachusetts. The author of numerous journal articles, Dr. Bridger's research focuses on constructive analysis, the philosophy of science, and the use of technology in mathematics education.

目录

Preface

Acknowledgements

Introduction

0 Preliminaries

0.1 The Natural Numbers

0.2 The Rationals

1 The Real Numbers and Completeness

1.0 Introduction

1.1 Interval Arithmetic

1.2 Families of Intersecting Intervals

1.3 Fine Families

1.4 Definition of the Reals

1.5 Real Number Arithmetic

1.6 Rational Approximations

1.7 Real Intervals and Completeness

1.8 Limits and Limiting Families

Appendix: The Goldbach Number and Trichotomy

2 An Inverse Function Theorem and its Application

2.0 Introduction

2.1 Functions and Inverses

2.2 An Inverse Function Theorem

2.3 The Exponential Function

2.4 Natural Logs and the Euler Number

3 LimitsSequences and Series

3.1 Sequences and Convergence

3.2 Limits of Functions

3.3 Series of Numbers

Appendix I: Some Properties of Exp and Log

Appendix 11: Rearrangements of Series

4 Uniform Continuity

4.1 Definitions and Elementary Properties

4.2 Limits and Extensions

Appendix I: Are there Non-Continuous Functions?

Appendix XI: Continuity of Double-Sided Inverses

Appendix III: The Goldbach Function

5 The Riemann Integral

5.1 Definition and Existence

5.2 Elementary Properties

5.3 Extensions and Improper Integrals

6 Differentiation

6.1 Definitions and Basic Properties

6.2 The Arithmetic of Differentiability

6.3 Two Important Theorems

6.4 Derivative Tools

6.5 Integral Tools

7 Sequences and Series of Functions

7.1 Sequences of Functions

7.2 Integrals and Derivatives of Sequences

7.3 Power Series

7.4 Taylor Series

7.5 The Periodic Functions

Appendix: Binomial Issues

8 The Complex Numbers and Fourier Series

8.0 Introduction

8.1 The Complex Numbers C

8.2 Complex Functions and Vectors

8.3 Fourier Series Theory

References

Index