Understanding Analysis (2nd Edition)
Quick Informations
Where to buy it: Springer
Author: Stephen Abbott
Release date: May 20, 2015
Language: English
Personal Progress: Read completely
Review
Description
Calculus is taught in an unrigorous way because in a calculus class, students are not just mathematics students but also engineers, physics students, and nearly any other natural science students. Hence, most definitions are ambiguous, most theorems are not stated clearly and most proofs are skipped. Understanding Analysis is perfect for fixing this problem. This book can be seen as a rigorous calculus textbook where everything is proved with a more than satisfying level of rigor. Moreover, with such a level of rigor, some new aspects of calculus arise that would never come to mind when first learning about calculus. This book shows why some of the basic concepts of calculus are actually far from being trivial.
Prerequisites
The most important prerequisite for this book is to already be familiar with calculus. Even if it builds everything from the beginning, this book really shines when you think that you know what continuity is and then see how wrong you were. Moreover, the book is also intended to be a first introduction to pure Mathematics, so it starts by introducing proof methods, basic logic and sets. However, it really helped me to have read A Transition to Advanced Mathematics before. Thus, having read a book on proofs would help a lot but it is not required at all.
Content
The structure of the book is, on the overall, very similar to a regular calculus class: Chapter 1 is on the Real Numbers, Chapter 2 is on Sequences, Limits and Series, Chapter 4 is on Continuity, Chapter 5 is on Derivatives and Chapter 7 is on Integrals. However, each chapter goes very deeply into the subject.
After reading the chapter on the Real Numbers, I realized that I never really knew what was the difference between the rationals and the reals beside the fact that there were reals that weren't rational (such as root 2). This feeling comes back at every single chapter (and even section) of this book. For example, Chapter 7 is on the Riemann Integral and I never realized before reading this chapter how complicated was the notion of integrability. In calculus, every function is integrable, in this chapter, the limit of integrability are explored until we find a clear criterion of integrability which seems to be more powerful than any theorem seen in any calculus class.
The last chapter (Chapter 8) is called Additional Topics. I felt like this chapter really was a playground to see how powerful were the tools we built in the seven previous chapters. The topics discussed were very interesting and sometimes very deep: The Generalized Riemann Integral, the Baire Category Theorem, a rigorous proof of the valuation of the sum of the reciprocals of the squares, the Gamma Function, Fourier Series and the construction of the Real Numbers with Dedekind Cuts.
Exercises
The exercises in this book are amazing and are a good complement to the discussions, explanations and given proofs. Some useful results are only presented in the exercises which makes some exercises hard to skip. After doing all of them, I really have the feeling that I understand very single sentence in this book and that I will never have to come back to this book again to recall a result. If I have time, I should probably write my solutions in LaTeX and put them in the Textbook Solutions page of this website.
Further Readings
Chapters 7 and 8 open on many different topics. The end of Chapter 7 introduces very briefly the notion of measures. A good follow up to this book would be a textbook on Measure Theory and Lebesgue Integration, such as Measure, Integration & Real Analysis by Axler. Similarly, one of the topics discussed in Chapter 8 is Fourier Series. A good follow-up that is more focused on Fourier Analysis is Fourier Analysis: An Introduction by Stein and Shakarchi. For the other topics discussed in Chapter 8, I would have to read more books to come up with some good recommendations, so I will stop here.