Real Analysis through Modern Infinitesimals

by Nader Vakil

Overview

Real Analysis Through Modern Infinitesimals provides a course on mathematical analysis based on Internal Set Theory (IST) introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the book provides a careful development of this theory representing each external class as a proper class. This foundational discussion, which is presented in the first two chapters, includes an account of the basic internal and external properties of the real number system as an entity within IST. In its remaining fourteen chapters, the book explores the consequences of the perspective offered by IST as a wide range of real analysis topics are surveyed. The topics thus developed begin with those usually discussed in an advanced undergraduate analysis course and gradually move to topics that are suitable for more advanced readers. This book may be used for reference, self-study, and as a source for advanced undergraduate or graduate courses.

Product Details

ISBN-13:	9781107002029
Publisher:	Cambridge University Press
Publication date:	02/17/2011
Series:	Encyclopedia of Mathematics and its Applications , #140
Edition description:	New Edition
Pages:	586
Product dimensions:	6.40(w) x 9.30(h) x 1.40(d)

About the Author

Nader Vakil is a Professor of Mathematics at Western Illinois University. He received his PhD from the University of Washington, Seattle, where he worked with Edwin Hewitt. His research interests centre on the foundation of mathematical analysis and applications of the theory of modern infinitesimals to topology and functional analysis.

Table of Contents

Preface; Introduction; Part I. Elements of Real Analysis: 1. Internal set theory; 2. The real number system; 3. Sequences and series; 4. The topology of R; 5. Limits and continuity; 6. Differentiation; 7. Integration; 8. Sequences and series of functions; 9. Infinite series; Part II. Elements of Abstract Analysis: 10. Point set topology; 11. Metric spaces; 12. Complete metric spaces; 13. Some applications of completeness; 14. Linear operators; 15. Differential calculus on Rn; 16. Function space topologies; Appendix A. Vector spaces; Appendix B. The b-adic representation of numbers; Appendix C. Finite, denumerable, and uncountable sets; Appendix D. The syntax of mathematical languages; References; Index.

Editorial Reviews

"The book is written in a very clear style, with many examples and exercises. On the whole, this book is commendable and will provide the reader with a clear introduction to analysis, and to the use of nonstandard analysis for learning analysis."
Antoine Delcroix, Mathematical Reviews

"Nader Vakil has shown with his text that advanced calculus and much of related abstract analysis can be explained and simplified within the context of internal set theory."
Peter Loeb, University of Illinois, SIAM Review

"Real Analysis through Modern Infinitesimals intends to be used and to be useful. Nonstandard methods are deployed alongside standard methods. The emphasis is on bringing all tools to bear on questions of analysis. The exercises are interesting and abundant."
James M. Henle and Michael G. Henle, MAA Reviews

From the Publisher