Fundamentals of Abstract Analysis

Every mathematics student’s aim should be to understand the mathematical ideas behind a chain of formalized propositions. The uniqueness of this book is that the author has succeeded in assisting the student to accomplish this.

The meticulous text of Foundations of Abstract Analysis is very abstract, formalistic, and based on naïve set theory. In the Preface, the author points out that the most “modern” mathematics books explain the axiomatic rather than the set-theoretic method. However, it is the later which has immensely contributed to our understanding of even the most primitive concepts. Thus, he has set out to “attempt to explain to the beginning student the relation of the set-theoretic mathematics to mathematics itself.” Gleason focuses on and clarifies the close relationship between “formalized” and “real” mathematics.

Even a simple read-through of the table of contents of Fundamentals of Abstract Analysis reveals a remarkable book. The first six chapters are devoted to elementary logic and set theory, while the three subsequent chapters discuss mathematical induction, fields, and the different “number systems,” from the natural integers to the complex numbers. Chapter 11 goes back to set theory, explaining countable sets, cardinal numbers and the axiom of choice. Finally, the book covers limits and arithmetic of complex sequences, finite sums and products, infinite series and products, topology of metric spaces, and the elementary theory of analytic (holomorphic) functions of one variable. (In the latter there is no integration theory and all the work is based on power series.) The explanations of standard mathematical and logical procedures are simple and clear.

The exercises at the end of each section have been carefully selected and organized in increasing order of difficulty. The ones marked with an asterisk require knowledge and techniques not included in the text. Solutions and hints to most of the exercises can be found at the end of the book. In addition, there are many propositions (especially in Chapter 14) whose proofs are left as exercises for the reader.

The book combines great exposition, rigorous treatment of the subject, and very well chosen problems. It can be recommended for undergraduate students with a deeper mathematical maturity than that gained by the beginner calculus courses. It is also an ideal choice for introductory courses in thinking, methodology, and abstraction of higher mathematics. It is one of my favorite books and an appropriate choice for anyone who is serious about the challenges and rewards of a full treatment of what for many is considered a difficult subject.

A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.