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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.Read more
- Will appeal to readers with no background in mathematical logic
- Emphasis on applications is ideal for readers seeking experience in applying modern infinitesimals
- Flexible teaching resource - instructors can apply the material to a wide variety of courses
Reviews & endorsements
"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 ReviewsSee more 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
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- Date Published: March 2011
- format: Hardback
- isbn: 9781107002029
- length: 586 pages
- dimensions: 234 x 156 x 32 mm
- weight: 0.99kg
- contains: 42 b/w illus. 1000 exercises
- availability: Available
Table of Contents
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
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
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