# An Overview of Selected Real Analysis Texts

# Highlights of some textbooks in real and complex analysis

Joshua Siktar

Real analysis, more than any other subject of math I’ve encountered, has been illuminating to me as to why **math is cool.**

At least at the basic level I’ve found the results of real analysis to be intuitive and the proofs elegant, even if said proofs were often far more than straightforward applications of the definitions.

This is in contrast to abstract algebra, where I found my first exposure to the subject to be overwhelming and unintuitive. That’s not to say it’s bad or unimportant, just that studying analysis relatively early on was more conducive to my desire to study more math, and in fact begin a PhD in mathematics.

The undergraduate-level real analysis sequence I took during my sophomore year at Carnegie Mellon laid a foundation for other courses I took later, including complex analysis and measure theory.

I will also be taking my first functional analysis course next semester at the University of Tennessee. Through all this coursework I’ve amassed a collection of books in these different facets of analysis, and wanted to share some of them.

I want to compare and contrast five of them, emphasizing what level of background is needed for each book. That way you have some guidance if you’re looking to add another book to your collection for self-study, or maybe even to teach a course in your home math department.

I’m not going to rank the books against each other because they all have slightly different audiences and approaches to writing about mathematics. The content also varies widely from book to book. Instead I’ll alphabetize them by the authors’ last names.

Abbott’s book is usually my go-to when somebody asks me for a recommendation for a first text in real-analysis. This book completely omits multivariable treatment and focuses purely on one-dimensional aspects of the subject, perhaps because most results are most easily visualized in one dimension.

Particularly in the early chapter on limits, this book uses a lot of visuals to gently guide its readers into what may be unfamiliar territory for them. Abbott decides to make some “classic” problems and anomalies in real analysis the centerpieces of the text, using these as motivators for the core concepts that are introduced thereafter.

Perhaps the most famous of these is the Cantor Set, which highlights some subtleties of the topology of the real line. There are also “special sections” near the end on Fourier Series and the Baire Category Theorem.

Other than that, the book is designed for use in a one-semester course in real analysis (I took such a course, and Abbott indicates in his preface that this was his intent in writing the book). I have mixed feelings about some of the exercises being main theorems that would proven in other texts.

While it may make students feel empowered to prove such important theorems on their own, I feel the exposition suffers on occasion due to this decision by the author. Of course, most of those “big” theorems have proofs lying around on Wikipedia or MathStackExchange.

We have here the first complex analysis text discussed in this article, the text by Bak and Newman aimed at undergraduates. First let me say this book is, in my opinion, suitable for a graduate-level course because the rigor and scope of topics parallels, if not exceeds, that of my graduate-level complex analysis course at the University of Tennessee.

The first two chapters are devoted to prerequisite material, and then the Cauchy-Riemann Equations and entire function theory are studied closely. Only then are the many variants of Cauchy’s Integral formula built up.

The cornerstone application of Cauchy’s Integral formula (and the closely related Residue Theorem) is the evaluation of real-valued definite integrals that cannot be handled with techniques from freshmen-year calculus.

This text is *not* shy with those by any means, devoting two entire (no pun intended) chapters to the topic. A few of the examples involve binomial coefficients, which gives this particular book the advantage of having direct appeal to those who study combinatorics.

Finally, there is a large range of exercises and I’ve even used this book as a supplementary resource as I prepare for my preliminary exams in graduate school.

I have also used this book as a supplement for my complex analysis course in graduate school. It works out a lot of examples which has certainly been helpful for studying.

The content is similar to Bak and Newman’s book but the order in which the content is presented is very different. In particular, the Residue Theorem is introduced much earlier, in a very hefty Chapter 2.

The entire last chapter is devoted to Fourier transforms, Laplace transforms, and their applications to differential equation theory. It seems that some familiarity with multivariable calculus is enough to handle this book, as use of rigorous concepts from real analysis is somewhat limited.

One little gem hidden in the back of the book (that I didn’t discover until nearly three years after purchasing the book) is an appendix full of conformal mappings.

These are “angle-preserving” functions, and are a common application of some of the fundamental building blocks of complex analysis. From my experience these conformal maps give a sense of urgency to being able to understand complex analysis from a visual perspective.

Many a good problem asked for an invertible conformal map between the unit disc and some other open subset of the complex plane. Hence the appendix will give examples that are quite useful as a reference or source of inspiration to tackle other problems.

Compared to some of the other books on my list, I suspect this one may have fallen off the radar a tiny bit, because it was originally published in 1968!

However, I don’t think that’s a reason to discredit the book because it’s delivered at the undergraduate real analysis level and could serve as a book for a one-semester (or first semester) course in the subject.

Such a course doesn’t generally rely on concepts developed in the past 50 years, but rather on concepts that are multiple centuries old.

Here’s the breakdown for this book’s contents in more depth. Preliminary chapters on set theory, the real number system, and metric spaces open the development for what’s considered some of the core material in one dimensional real analysis: continuity, differentiability, and interoperability.

The treatment of limits is built into the metric spaces chapter. Chapter 7 includes theory on infinite series, rather than integrating this material into other chapters or omitting it entirely.

The final chapters include some more advanced material penetrating into multivariable real analysis, including, surprisingly enough, some treatment of existence theory for differential equations.

I know there is some debate about whether or not it is advisable to introduce abstract metric spaces in a first real analysis course.

Personally I don’t think it adds very much abstraction such that its introduction should be delayed to later courses, especially because many practical applications of the theory wind up involving the Euclidean metric anyway.

For those instructors who wish to only work with the Euclidean metric, this book might be a bit difficult to use as the main text for the course.

One of the highlights of this book is its direct derivation of the sine and cosine functions without using any geometry. It instead relies on Taylor Expansions, which are made rigorous through the one-dimensional differentiation theory in the first half of the book.

I found this development to be so much more clear than the traditional one, and the return to [presumably] familiar ground is likely encouraging for many students who might otherwise get lost in the weeds of technical proofs.

For the grand finale we have what is arguably the most advanced book in this list. The title is a bit of a misnomer since this text is at the graduate level, compared to the undergraduate level of many other books with “Real Analysis” somewhere in the title (including Abbott’s book).

Speaking of which, I’d recommend having a semester of undergraduate real analysis *before* picking up this book. Even if that background knowledge isn’t strictly necessary, many of the concepts are repeated in a much more abstract setting, sometimes multiple times over.

Furthermore, the pace of this book is quite fast, and it would likely take three semesters to cover all of the book’s contents thoroughly. In particular, the first eight chapters roughly correspond to a standard first course in measure theory (often taken in one’s first semester of a mathematics graduate degree program), but some topics are deferred to near the end of the book.

Later chapters quickly introduce topological notions that could be used in a separate topology course, and then Banach and Hilbert spaces are studied at some length. Weirdly enough, Lebesgue measure in arbitrarily many dimensions doesn’t find its way into the book until Chapter 20.

Throughout the book, Royden seems to emphasize treating theorems very thoroughly, and he includes many exercises devoted to challenging the reader to evaluate what happens if certain conditions are added to or removed from the theorem statement.

Thus I think it’s a great book for deepening one’s understanding of graduate-level analysis. Unless, of course, you want to study complex analysis, which is pretty much abandoned in this text. Luckily I’ve already given you a few suitable alternatives for that.

I’d like to comment that while real and complex analysis are beautiful subjects on their own and some mathematicians find a lifetime of enjoyment just from studying these two subjects in depth, they are invaluable in applications to other fields of mathematics.

One of the most closely studied such areas is that of partial differential equations (PDE), and it’s one of my personal favorites. My collection of books extends into this subject as well, and I wrote an accompanying article in the same format about PDE books.

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Joshua Siktar

Ph.D. Candidate, Applied Mathematics, University of Tennessee-Knoxville | B.S. Mathematics, Carnegie Mellon | Facilitator of Modernization of Education

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