Description
This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.
Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.
I found this book very easy to understand and written thoughtfully. I was able to understand Real Analysis thanks to it. It is not daunting like the other textbooks and filled with a lot of context that helps the student more easily familiarize themselves with the subject matter. I mean, imagine that being your first experience with Real Analysis!!! I am Impressed.
According to John Derbyshire. Mathematics is traditionally divided into four subdisciplines: arithmetic, geometry, algebra, and analysis. You know what arithmetic and geometry are, and you probably have taken a high-school algebra class. “Analysis”, however, is a little obscure. The word has a specialized meaning in mathematics. It is that branch of mathematics that includes calculus. More properly, analysis is the mathematics of the continuum.
The calculus was developed in the late 17th century by Isaac Newton and Gottfried Leibniz. Newton and Leibniz however, didn’t quite know what they were doing and inevitably they were a little sloppy about defining things. (This is usual when a new area of mathematics is developed.) At the heart of the problem was this: calculus is all about continuous things — in calculus space and time are continuous. What that means roughly is that we assume in calculus that every point on the line between two points A and B exists. (There is reason to believe this may not be physically true, but that is not relevant to the mathematics under discussion.) Furthermore, we assume that a number can be assigned to every one of those infinity of points.
That is not a precise definition of continuity. Defining continuity is surprisingly difficult. The ancient Greeks were aware of the problem — this is what Zeno’s paradoxes are all about. Furthermore, the Greeks knew that no number (as they understood numbers) could be assigned to the length of the diagonal of a 1 ⨉ 1 square.
In the 19th century this problem was figured out by European (mostly French and German) mathematicians. Some names to conjure with here are Weierstrass, Dedekind, Cauchy, Riemann, and Cantor. These are names every mathematician knows. Over the course of several decades they figured out how to rigorously define the continuum and to assign a number to every point on the line. These are called the real numbers, symbolized ℝ. The 19th century analysts did work of astonishing beauty, which, sadly, most people will never perceive. Analysis is now a course that every undergraduate math major is expected to take. It is generally regarded as the most difficult such math class.
In 2015, I was a professor with a 40-year career as a scientist behind me. I decided to retire and go back to school for an advanced degree in mathematics. I had never taken a course in analysis. That was a gap in my education I needed to remedy. I therefore worked my way carefully through Stephen Abbott’s Understanding Analysis. This worked. In fall 2015 I took my first actual analysis course — Functional Analysis, a postgraduate course. I don’t remember my exact grade, but it was in the 90s.
So that was good — it was why I read Abbott — I got what I hoped from it. But I got much more than that. I was not prepared for the aesthetic experience. Math students don’t talk about the beauty of analysis — generally they are too traumatized by the effort to get through the most difficult course they have ever taken. Abbott does, though. In his preface he writes,
“Yes these are challenging arguments but they are also beautiful ideas. Returning to the thesis of this text, it is my conviction that encounters with results like these make the task of learning analysis less daunting and more meaningful.”
So, I will dare to challenge Edna St. Vincent Millay — it is not Euclid alone who has seen beauty bare. Weierstrass, Dedekind, Cauchy, Riemann, and Cantor have also seen her. And thanks to Abbott, I, too am one of those fortunate ones
“Who, though once only and then but far away,
Have heard her massive sandal set on stone.”
Pedagogically, I found this book wonderful. The way the material is exposed is insightful, and the book has proper typesetting. For the motivated reader (this is maths after all), it’s a pleasant read.
One thing I’m not too happy about is the binding quality. I am usually pretty meticulous and careful with my books, and normally hardcover books of this type are very tough, but by the time I was done reading the first two chapters, the binding was broken in several places. Roughly half the pages seem to be hanging by a thread. This has never happened to me with any of the other Springer hardcover books in this format. My product shipped from India for some reason, so although it doesn’t look counterfeit, a part of me wonders if I purchased the real thing. It’s a crazy world we live in, who knows.
I like that this book covers minimal topics and not fixating on the very fundamentals of analysis(not that it is unimportant but those are the barriers that make the students stumble), but I didn’t like the way that the author exposes the topics and proofs. Very often the sentences or expressions that the author added during the proof causes more confusions. It would be much nicer if the author gave a clear proof and then explains some motivations or guidance for the proof.
Amazing text, but almost half of the book is just left as tricky exercises. Not recommend for self studying, unless your papa is called Rudin.