Principles of Mathematical Analysis

It's hard to know where to start with “baby Rudin.” There is probably no more well known, respected, loved, hated, and feared text in all of mathematical academia. Steven G. Krantz claims it’s one of the books that made him a mathematician; “It was easy to say, and often true, that anyone who could survive a year of Rudin was a real mathematician.” Vladimir Arnold, one of the champions of the “organic” approach to mathematics, reportedly called the book (in comparison to the lectures of Vladimir Zorich) “Bourbakian propaganda, stripping and sterilizing analysis of any soul or meaning beyond the symbols.”

At MIT, the book has been practically canonized: I was once visited by some of my friends taking math in Cambridge and I was angrily dismissed as an ignorant dabbler for even suggesting any other text for undergraduate real analysis even existed. On the other hand, there was a group of math and physics majors at NYU who bought 100 copies of the book merely to burn the entire pile as a statement of their contempt for it.

Love it or hate it, the book elicits incredibly strong passions in people. It also remains the single most assigned text for undergraduate real analysis by professors. Hence, despite all that’s been written on the book, I’d like to put my 2 cents in as far as “The Blue Book” is concerned.

It’s helpful to put the book in a historical perspective, since very few students today know how the first edition of the book came to be written. Oral tradition — as well as Rudin’s own account, in his The Way I Remember It — holds the book had its genesis in an advanced calculus/elementary real analysis course given by Rudin when he was a young C. L. E. Moore Instructor at MIT in the early 1950s.

Rudin was discussing the difficulty of choosing a suitable text with Ted Martin, then chair of the mathematics department at MIT. Back then, there simply were no modern texts on classical real analysis in English. There were the old British classics such as Hardy’s Pure Mathematics and Whittaker and Watson’s Course of Modern Analysis, but these were written in a late 19th century style that made no mention of topological concepts. There were modern treatises on real variables, such as Hans Hahn’s books Theorie der reellen Funktionen (1921), and Reelle Funktionen (1932) — but these were not available in English. More importantly, they were too advanced for such a course. Martin quite naturally suggested Rudin write such a text. After several versions of the course and the resulting lecture notes, the first edition of Principles was published in 1953 and the rest was history — for better or worse.

It’s important to note in retrospect that Rudin’s text was the first of its kind. The list of topics in the book has, as for any text that was both the first of its kind and a great success, become almost clichéd. Chapter 1 gives a detailed study of the real number field. Earlier editions used the direct construction of R via Dedekind cuts of rationals. In this edition, Rudin relegates the construction to an appendix, arguing that this construction is too difficult for beginning students. Instead, he introduces the reals axiomatically using the Archimedian, completeness, order, and field properties. While it may be more indirect, this method is overall a little easier for students to digest, particularly those with some training in set theory or algebra.

This chapter also gives some very deep insights into how R differs from other fields, such as the rationals. These results can be hard to find in other analysis books. A particularly nice example is Rudin’s detailed analysis of why the equation \(p^2=2\) has no solution in the rationals, demonstrating that the greatest lower bound/least upper bound principles do not hold for Q.

Chapter 2, taken with its exercises, gives a very complete account of the topological properties of R as a metric space. Chapter 3, on numerical sequences and series, is, to me, the best chapter in the book. There’s a very good discussion of sequences and series, Cauchy and power series with their associated convergence tests and summation formulas, with a ton of examples and counterexamples that are hard to find in other books at this level. This chapter also pretty much sets the tone for the rest of the book: It is crystal clear but concise to a brutal degree. Proofs and most examples are stated brusquely with little or no further explanation beyond the definition and the statement of fact.

Chapter 4 discusses continuous functions in metric spaces, the relation between connectedness and continuity in metric spaces, compactness in such spaces, discontinuities, infinite limits and uniform continuity. Chapter 5 discusses differentiation of real valued maps in general metric spaces — the highlight here is a careful proof of the general L’Hôpital limit theorem that is taken for granted in even the best calculus texts. Chapter 6 gives a presentation of the Riemann-Stieltjes integral defining partitions, upper and lower R-S sums over closed and bounded subsets of R and their “convergence theorems” as well as the major theorems of integration in calculus.

Chapter 7 covers sequences and series of functions and their convergence properties, focusing on uniform convergence as the central notion for developing the properties of function spaces of real valued maps, such as equicontinuity and the Stone-Weierstrass theorem. Chapter 8 uses these properties to develop some useful functions of real analysis, such as power series, Fourier series and the Gamma function.

The final three chapters are nowhere near as good as the previous 8: Chapters 9 and 10 develop multivariable calculus in a rigorous fashion using linear maps and differential forms over Rⁿ. I think these are by far the weakest chapters of the book The reason is that a rigorous, concise presentation of this material in keeping with the aims of the book requires far more topology and linear/multilinear algebra than the crash course Rudin provides. The development he does give is strictly what he needs to be able to prove the Implicit and Inverse Function theorems in Chapter 9 and the general Stokes theorem in Chapter 10 — the matrix algebra, linear algebra and vector space theory is given on a “need to know” basis. The result is more confusing than informative.

The book concludes with a quick overview of the Lebesgue integral in chapter 11, which seems tacked on and forced. Measure theory may be the single most important subject any student studying analysis will learn — what good will this mixed bag “sampler” chapter do him or her? It seems to be there largely for the sake of completeness to make the book as “modern” as possible.

Prior to chapter 9, however, the book gives a masterly presentation of real analysis for the serious math student. The book is brutally concise; more then half the “proofs” in the book amount to little more then hints. Most fans of this book will tell you, however, that spoon feeding mathematics is only harming the development of budding mathematics majors — only by banging your head against substantial problems will a math major truly learn the way of proof. It’s hard for any experienced math student to argue with this “tough love”assessment. Despite its conciseness, Rudin does for the most part supply enough detail that able students can fill in the blanks.

One does wish Rudin had included more examples in the presentation. I’m a firm believer that four or five well chosen examples are more instructive to a developing math student then the most detailed proofs. The greatest strength of the book, however, are the legendary problem sets at the end of each section: Most are long, difficult tasks that will impress many deep concepts on the student. Surprisingly, Rudin usually does give excellent hints pointing the student in the right direction. Indeed, in some ways doing the exercises is more pleasant then reading the actual text!

So — does the book deserve its classic status? My answer: Yes, but that’s not an unconditional yes. The question that needs to be asked first is what kind of mathematics background do the prospective students have? The reason this question needs to be asked first is because the "average" answer has changed dramatically since Rudin wrote the first edition of this book over half a century ago.

It’s easy to look at this book as a mathematician and regard it with reverence — hindsight is 20/20. But as an undergraduate in one of today’s colleges, struggling with many of the counterintuitive aspects of rigorous calculus for the first time, is this book the first choice?

The answer appears to me to be yes if the students: (a) Understand and have some experience doing proofs, as would be obtained in an honors calculus or non-applied linear algebra course. (b) Have some experience working with inequalities and approximation arguments and not merely what my graduate advisor Nick Metas likes to call “pencil pushing” mathematics, i.e. exact solution algebra problems which are virtually useless in the real world.

Requisite (a) is fairly easy to supply in today’s university environment if one looks hard enough — even if only weak calculus is available as preparation, a good linear algebra course is usually readily available. Unfortunately, the second condition has become nearly impossible to find at all but the strongest of programs.

Calculus courses in the USA have been transformed from strong mathematical crucibles, in which approximation and geometrical proofs were part and parcel of the subject, into much less rigorous courses taken by all or most incoming freshman science majors. When Rudin wrote this book, calculus courses included epsilon-delta limit arguments and inequalities on the real line alongside related rates, solving differential equations and calculating volumes and areas using standard integral formulas. Looking at the books of the past — such as Lipman Bers’ Calculus and Edwin E. Moise’s Calculus — it’s easy to see why Rudin was the book of choice for analysis courses. It was reasonable to expect that students who did well in such calculus courses would have more then sufficient background to be able to tackle Rudin, despite the effort it would require of even good students.

Today’s students don’t stand a chance — most are simply overwhelmed due to lack of preparation. It’s as simple as that. Unless they’ve had the good fortune and talent to be guided through high school to a good honors calculus course as freshmen — such as those based on Spivak’s Calculus — reading this book is going to be a real struggle, to say nothing of the exercises.

My argument, then, is that this book is a classic, but it needs to come with a warning: For Mathematically Mature Audiences Only. Fortunately, the need for a "bridge to Rudin" for students with insufficient backgrounds has been recognized by a number of educationally-minded mathematicians in the last 30 years. It seems appropriate to mention some of these in closing. Kenneth Ross’ Elementary Analysis: The Theory of Calculus i s a classic in it’s own right, developing material that simply cannot be found anywhere else with a host of carefully solved “epsilon” problems on the real line. It’s exactly what the title says it is, a student that masters it will be well prepared for a follow up course based on Rudin. R. P. Burns’ wonderful Numbers And Functions gives a gentle theoretical calculus course in the form of a problem course with carefully given step by step hints — this would make terrific summer study preparation for a Rudin based analysis course in the fall.

Ideally, though, one wouldn’t want to give a student a “remedial” third term to prepare for a year long analysis course. That’s why my real analysis book of choice is the second edition of Steven G. Krantz’s outstanding Real Analysis And Foundations. The book covers all the same topics as Rudin’s and the influence of the book on Krantz is clear. But there is far more explanation, examples and details and those are chosen with the greatest of care so the student is not spoon-fed. There is also a set of wonderful, tough exercises. Best of all, the book’s level of difficulty gradually increases towards the advanced topics at the end. This is a modern Rudin well suited for today’s students: It educates them at a level appropriate for their preparation, but it still succeeds by the end in giving the student a deep, challenging course in the wondrous world of real analysis.

Andrew Locascio is currently a graduate student at Queens College Of The City University Of New York.