Fundamentals of Abstract Analysis 1st Edition by Andrew Gleason (PDF)

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This classic is an ideal introduction for students into the methodology and thinking of higher mathematics. It covers material not usually taught in the more technically-oriented introductory classes and will give students a well-rounded foundation for future studies.

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Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Note: Solutions to most exercises concludes this textbook (1991 edition). The solutions consume more than fifty pages (thus, a useful reference). Preface: exercises are arranged in order of increasing difficulty. Excellent pedagogy ! If the solution is not provided, a hint serves in its stead . The preface: “…intended for…fourth-year students, give, or take a year.” A third-year student might find the text too challenging. Now, it seems to me that many an instructor would eschew this text due to its idiosyncratic and creative discourse (no integration theory, though). Hopefully, I am in error on that score, as this is an insightful textbook:(1) Sets, Logic, Functions and Relations: fundamental concepts for which you will spend the initial fifty pages. Abstract, yet necessary foundation.(2) We note: ” an ordered pair will be called a configuration.” (page 55). Read: ” set-theoretic descriptions are often clumsy, oblique, and non-intuitive. Still, they are always precise.” (page 61). Style of the author: glance at the proof of proposition #5.1.3 (page 66); short, but sweet !(3) Chapter Six is where we really hit the ground running: Order…”is an abstraction of the notion of dominance.” Careful reading of this chapter will serve one well. Next, a favorite topic of mine:(4) Mathematical Induction: a ten page exposition. The concept of “chains” introduced. (Read: “a configuration of a set and a function.”)(5) Read: “The structure we call the real number system is to be an idealization of the numbers we use to measure quantity in the physical world.” This in eighth chapter on Fields. Note, the proof which “typifies the use of fixed-point theorems.” (page 109). Real Numbers ‘constructed’ in chapter nine where “it provides an excellent demonstration of set-theoretic techniques.” (pages 112-128).(6) Brief interlude (six pages) introduces complex numbers. More about them later(chapter fifteen). Another pedagogic highlight: the interesting discussion of “another form of induction” (Section 11-4, pages 144-148) which segues to ‘the axiom of choice.'(7) Chapter Twelve: limits. Begin with sequences (for real and complex numbers). A nice exposition of limits and their arithmetic intervenes: with a nice proof (pages 170-171) of proposition #12-2.2. Read: “Most of the skill in finding mathematical proofs consists of choosing appropriate specializations of universally quantified propositions.”Read: ” we develop criteria for the existence of limits. These are consequences of the completeness of the real numbers.” (page 180). Bolzano-Weierstrass theorem concludes the chapter.(8) Next, Infinite Series (page 191): “Most classical books define an infinite series to be a notation, not a mathematical object.” and “the distinction between an infinite series and its sum is important, so it seems worthwhile to distinguish them in the notation.” Excellent pedagogy ! We get an excellent discussion of the various tests for convergence, even a brief discussion of infinite products (page 215). We read: “the situation becomes transparent with the aid of logarithms….since log is an order-preserving bijection from the positive reals to R…”(9) Topology of Metric Spaces: the discussion segues to continuous functions (page 239) then onward to uniform continuity (page 245). Homeomorphism is defined. (page 249). Compact spaces “resemble in many ways the properties of finite sets.” Learn of them, page 266.(10) Chapter fifteen revisits complex functions: sine, cosine, exponential via Taylor (power) series (pages 308-315). Note: there is no integration and thirteen pages of geometric applications concludes the chapter. That is, “we shall prove some theorems which connect the theory of analytic functions to the geometry of the plane.” (page 326).(11) Summarizing: A well written account of fundamental principles generally applied to the term ” mathematical analysis.” The exposition is lively, the proofs lucid, the exercises both straightforward and challenging.A minor quibble: no references or bibliography. The subject index is adequate (seven pages). Keep in mind, this is not “re-doing calculus” at a more abstract vantage–you look elsewhere for that. Highly recommended, especially as a reference work and a source of inspiration.

⭐Recommended for mathematicians who really want to have a better understanding of the foundations of mathematics! Magnificent work!

⭐Fundamentals of Abstract Analysis was our prescribed text in my Advanced Calculus class. I got hold of this book from our college library. The way Gleason presented topics form this course was brilliant. Although this course is not a basic calculus course, this is a must for mathematics majors who wish to pursue masters and doctorate degrees in the future. This book is recommended for students with a more mathematical maturity than that afforded by the usual freshman-sophomore courses in calculus. And one excellent way to gain such maturity is through a beginning course in linear algebra, although it is not a pre-requisite for the book, but of an advantage. This book tackles the “structure” of calculus, wherein theorems are presented in much greater depths and goes “behind the scenes” of the theorems in calculus. I recommend this book for undergraduate math majors and graduates as well.

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