Real analysis through modern infinitesimals
By: Nader Vakil
Series: Encyclopedia of mathematics and its applications -- 140
Publisher: Cambridge University Press
Published: 2011
Place of publication: Cambridge; New York
Language: eng
Number of pages: 565
ISBN-13: 9781107002029
Copyright date: 2011
Subjects: Set theory; MATHEMATICS / Mathematical Analysis; Mathematical analysis
Description: "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"--

"This book provides a course in mathematical analysis using the methods of modern infinitesimals, which are developed within the framework of internal set theory (IST), introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the author provides a careful development of the theory in which each external class is represented as a proper class. The basic standard and nonstandard properties of the real numbers follow, together with a thorough discussion of the central topics of analysis that begins with those usually discussed in an advanced undergraduate course and gradually moves to topics suitable for more advanced readers"--
Notes: Includes bibliographical references (pages 554-556) and index.
Table of contents: title: Machine generated contents note: 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; A. Vector spaces; B. The b-adic representation of numbers; C. Finite, denumerable, and uncountable sets; D. The syntax of mathematical languages; References; Index.