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Real Analysis : a Constructive Approach.

Author: Mark Bridger
Publisher: Hoboken : Wiley, 2012.
Series: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts.
Edition/Format:   Print book : Document   Computer File : EnglishView all editions and formats
Summary:
A unique approach to analysis that lets you apply mathematics across a range of subjects 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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Bridger, Mark.
Real Analysis : A Constructive Approach.
Hoboken : Wiley, ©2012
Material Type: Document
Document Type: Book, Computer File
All Authors / Contributors: Mark Bridger
ISBN: 9781118367759 1118367758
OCLC Number: 889674280
Notes: Includes index.
Description: 1 online resource (452 pages).
Contents: Cover; Contents; Title Page; Copyright; 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 Theorem2.3 The Exponential Function; 2.4 Natural Logs and the Euler Number e; 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 Ill: The Goldbach Function; 5. THE RIEMANN INTEGRAL; 5.1 Definition and Existence. 5.2 Elementary Properties5.3 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; 8.2 Complex Functions and Vectors; 8.3 Fourier Series Theory; REFERENCES.
Series Title: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts.

Abstract:

A unique approach to analysis that lets you apply mathematics across a range of subjects 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 a.

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