This is a showcase of how the methods of nonstandard analysis can be applied to advanced areas of real and abstract analysis (it is not about applications to real-world problems). The present volume is a Dover 2005 reprint of the Wiley 1977 original, with a new preface and with errata bound in.
Nonstandard analysis has its foundations in mathematical logic (specifically model theory), so the book begins with 40 pages laying the foundations. This portion is concise but very clear, and ends with a handy recapitulation (pp. 41–42) of the things that it is important to remember in the remainder of the book. The book then develops the real and hyperreal numbers and goes on to prove various theorems on topological and metric spaces, normed linear spaces, and Hilbert spaces.
The book makes heavy use of the transfer principle, which allows us to prove a result in either the standard or nonstandard universe (whichever is easier) and then transfer to the other universe. For calculus use we usually prove the result in the hyperreals (where the infinitesimals live) and then conclude it is also true for the reals.
Being a showcase, the book is not comprehensive. It tries to present some of the more interesting results and to bring out why nonstandard analysis is useful. It is thus very different from most nonstandard analysis books that focus on calculus and how nonstandard analysis has rehabilitated the infinitesimal (for example, Keisler’s Foundations of Infinitesimal Calculus and Henle & Kleinberg’s Infinitesimal Calculus). The book points out on p. 1 that “the subject can be claimed to be of importance insofar as it leads to simpler, more accessible exposition, or (more important) to mathematical discoveries.” An example of a theorem discovered through nonstandard analysis is the Bernstein--Robinson theorem on infinite-dimensional vector spaces (covered here on pp. 140–150).
Bottom line: an interesting and well-written book, but very specialized.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.