Real Analysis: A Constructive Approach

A unique approach to analysis that lets you apply mathematicsacross a range of subjects

This innovative text sets forth a thoroughly rigorous modernaccount of the theoretical underpinnings of calculus: continuity,differentiability, and convergence. Using a constructive approach,every proof of every result is direct and ultimatelycomputationally verifiable. In particular, existence is neverestablished by showing that the assumption of non–existence leadsto a contradiction. The ultimate consequence of this method is thatit makes sense not just to math majors but also to studentsfrom all branches of the sciences.

The text begins with a construction of the real numbersbeginning with the rationals, using interval arithmetic. Thisintroduces readers to the reasoning and proof–writing skillsnecessary for doing and communicating mathematics, and it sets thefoundation for the rest of the text, which includes:

Early use of the Completeness Theorem to prove a helpful InverseFunction Theorem

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

Emphasis on uniform continuity and its consequences, such asboundedness and the extension of uniformly continuous functionsfrom dense subsets

Construction of the Riemann integral for functions uniformlycontinuous on an interval, and its extension to improperintegrals

Differentiation, emphasizing the derivative as a function ratherthan a pointwise limit

Properties of sequences and series of continuous anddifferentiable functions

Fourier series and an introduction to more advanced ideas infunctional analysis

Examples throughout the text demonstrate the application of newconcepts. Readers can test their own skills with problems andprojects ranging in difficulty from basic to challenging.

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

Preface

Acknowledgements

Introduction

0 Preliminaries

0.1 The Natural Numbers

0.2 The Rationals

1 The Real Numbers and Completeness

1.0 Introduction

1.2 Interval Arithmetic

1.3Fine Families

1.4Definition 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

3 Limits, Sequences 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 II: 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 II: 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 and Functions

7.2 Integrals and Derivatives o 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

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.