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Real Analysis through Modern Infinitesimals

Nader Vakil
Cambridge University Press
Publication Date: 
Number of Pages: 
Encyclopedia of Mathematics and Its Applications 140
[Reviewed by
Leon Harkleroad
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Real Analysis through Modern Infinitesimals adds to the modest number of textbooks on nonstandard analysis that are aimed at an audience of upper-division undergraduates to graduates. The first part of the book, Elements of Real Analysis, addresses single-variable calculus — limits, continuity, derivatives, integrals, sequences, and series. The second part, Elements of Abstract Analysis, covers more advanced topics — for example, Banach spaces, the Daniell integral, the Baire Category Theorem, and integral operators.

The book approaches nonstandard analysis through Edward Nelson’s internal set theory. This axiomatic approach allows one to work with a hyperreal system without having to deal with its explicit construction in terms of ultrapowers and the like. However, Vakil actually presents the ultrapower construction first and then uses it as motivation for internal set theory.

Although the Preface states that “Our goal here is to explore the applications of modern infinitesimals in studying the central topics of real analysis,” the book treats nonstandard techniques as useful additions to the usual toolkit, rather than insisting on viewing everything exclusively through a nonstandard lens. Indeed, three pages later in the Preface, Vakil says that “[i]nfinitesimals are served as a side dish” and makes a point of mentioning that he introduces, e.g., continuity and the Riemann integral in delta-epsilon, rather than nonstandard, terms.

Real Analysis through Modern Infinitesimals contains a large (four-digit) number of exercises, both incorporated into the text and gathered into separate sections. Answers are not included.

In recent years Leon Harkleroad has mostly concentrated on mathematical aspects of music, but he still enjoys revisiting his old stomping grounds of mathematical logic.

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