Also known as Rudin’s Infamous Tiny Torture Box.
Trigger Warning: Baby Rudin.
I.
For my undergrad, I majored in Applied Mathematics at a university that was fairly well known for its mathematics department. One of the constants in “Applied Mathematics” is the use of calculus for a variety of applications. “The real world”. In statistics and machine learning, you might be trying to maximize a likelihood function to obtain parameters that describe your data. In economics, you might be trying maximize your ever-coveted utils. I have no idea what is maximized or minimized in real physics, so I won’t pretend to know, but I do know that at least in Newtonian physics, calculus is good at describing physical phenomena, since I am pretty sure people have exams that involve throwing balls and taking derivatives.
Since I was an Applied Mathematics major, I not only needed to know calculus, I needed to know the proofs behind calculus. Real Analysis is the study of the real number line from a mathematical point of view. I.e., limits, differentiation, integration. I.e., Calculus. I.e., Calculus where all you do is proofs. This implies that Real Analysis was a major milestone of my (and others) undergraduate mathematics career. As one of my favorite professors once said: “This class is serious. You’re all adults now.” Real Analysis is a trial that every mathematics major goes through in some form, and because it lays the foundations for much of the applied sciences, Physicists, Statisticians, and Economists often go through it as well. I think once in a while, even a computer scientist will take the course.
Analysis can come in many different flavors. Elementary Real Analysis usually refers to Real Analysis without the concept of a measure (as related to this blog’s title), and Elementary Real Analysis is usually what makes up the content of an undergraduate course. There are many, many books that teach Real Analysis, and many diverge in philosophy. Conversely, many of these books seem to converge to the same few topics in the same order.
The Real Line. Metric Spaces. Sequences. Continuity, Differentiation, Integration, Sequences of Functions. These topics in this order are practically canonical, and while some books may have a permutation of the above list, the list itself remains the same. No small part of that is due to a now infamous “introductory” book on Mathematical Analysis. A book that has earned the name “Baby Rudin” for being the smallest of the books published by the late Walter Rudin on the art and theory of Mathematical Analysis.
This isn’t a review on the book. This is a review on what it was like to learn analysis from the book.
II.
In hindsight, I was thrown into analysis perhaps a bit too quickly, and I struggled greatly with the subject. I went from non-rigorous calculus to a single 10 week proofs course to Baby Rudin. “A talented high schooler could handle Baby Rudin” you might think. Well it just so happens that I am not and was not a talented high schooler.
Rudin’s first two chapters seemed impenetrable. Looking back now, it wa straightforward, but there is a sudden jump in abstraction when going from calculus to real analysis. The limit of a sequence is something tangible, just a bunch of points getting closer together. A continuous function is just “a line you can draw without lifting your pencil”. A differentiable function is simply “smooth”. But these ideas almost seem to be thrown out the window and instead you discuss things like supremums and infimums. Metric Spaces. What even is a metric space? If you ask this question you might be thrown examples which don’t seem to answer the question. “Consider the trivial metric space in which every point has a distance of 0 with itself, or a distance of 1 with every other point. It satisfies the criterion for a metric space.” Whoa man, slow down, I’m still on the triangle inequality.
“I have to teach myself all this stuff? Why does no one explain anything? Why is the class so fast paced?” Part of what makes mathematics difficult is not just the level of abstraction and the true difficulty of logical, coherent thought, but the pace of the material. People who are “good at math” aren’t simply good at math. They digest new information quickly, and can feel logic almost intuitively. After an exam, I remember being told by a fellow class mate – “I have no idea what question 7 was about. But I knew that my answer was correct.” He’s taking graduate courses now.
It’s not an accident. Part of developing mathematical maturity is learning how to swallow abstract definitions and theorems and provide proofs for things with no intuitive understanding of the objects in question. The idea is that one learns how to thinking completely logically, rigorously. Knowing what an object “really is” is not required, and might even be discouraged when one is learning the ins and outs of rigor. You might argue that intuition is crucial to the developing mathematician, but why are you so sure? Rudin specifically demands a kind of abstracted thinking – there are no pictures, there are no explanations on “how” to think about anything. There are just definitions, theorems, and proofs, and a couple of examples of special cases. What you are learning is not what continuity is, but how to prove things with it, and about it, while not understanding it. The ignorance of the objects here is not a bug, it’s a feature. Over time, as one works with these objects repeatedly, one can determine their own way of thinking about things. I remember asking a friend of mine who was far more mathematically developed than I was for help on how to understand some Linear Algebra theorems. He shook his head and politely refused. “I have my own way of thinking about it. It’s hard to explain.” Indeed, math is hard to explain.
While every new, difficult course presents some growing pains, there really is nothing like seeing actual rigorous mathematical analysis for the first time. The rumor I’ve heard is that when Rudin wrote his little book, it was actually a set of lecture notes that were intended to bridge the gap between Calculus and serious mathematics. But “Calculus” as we know it now is nothing like Calculus “back then”. The rumor goes that when Rudin first started teaching out of his book, the students in his class were all pure mathematics majors, and had been doing delta-epsilon proofs already for a year, were familiar with logical statements and their proofs, and had some grasp of real line topology. So the abstraction of the real line topology to the metric space topology was no big deal. The ominous, unfamiliar statement “Consider the trivial metric space…” was taken in stride.
Meanwhile, you have no idea with an infimum is and your professor starts every sentence with Let or (if he is cultured) Suppose that…
“Why do I put up with this? How do they get away with it? We don’t learn calculus like that any more, and in fact I’m an engineering student! Our teacher is terrible and doesn’t explain anything!” You’re being hazed. Our similarity scores are probably closer than you think, but the real answer is that Rudin is going to teach you everything you need to know, while simultaneously not being intended as a lesson in metric space topology, sequences of functions, or continuity. While you will learn those things, somewhat, doing Rudin is more about teaching you how to be alone in a room with yourself, how to discover “tricks”, and how to prove properties about things while not understanding them. Your professor is not actually a teacher, he is more of a guide that doesn’t speak the native tongue and the map is Rudin. It’s really up to you to discover what is buried beneath it’s terse exposition.
III.
It can be pretty difficult to read mathematics. Personally, I can’t do it the way some of my friends can, which is where they literally read the book like a novel and somehow glean the information they need. I suspect they have a higher IQ than I do, or at least less traits of ADHD. I don’t know how to check this, but if I had to guess, my friends are outliers in the distribution and the rest of us unfortunate souls are a little closer to center.
Really, this is fine. Part of developing mathematical maturity is developing your own way of learning it, and thinking about it. The point of mathematical maturity isn’t to memorize all the definitions and theorems so you can call on them whenever you need to, but to learn how to learn math. Everyone has their own way, but I would argue that there are some very general ways to learn math that work for most people. Paul Halmos said something like “Don’t just read it, fight it!”, but what on Earth does that even mean?
There is a way to read Rudin that probably works for most people and I think this is it. Certainly, I can not take credit for this idea, which originally came from a rather hilarious amazon review for Rudin, but I did try it and it did work, for the most part.
It goes something like this. “Open the book at page one. Read until you come to the word ‘theorem’. Write the statement of the theorem down on a piece of paper. Don’t read the proof. Now close the book. Prove the theorem. At least, try to prove the theorem. You will get stuck. Open the book and read the first line of the proof related to where you are stuck. Close the book and continue from there. Repeat until the theorem is eventually proved. Repeat this process for the first 8 chapters. Enjoy grad school.”
Here is a slight modification that can help if the above is too hard: Read the proof once, quickly. Close the book and then reproduce it in full detail, omitting no steps, no matter how trivial. This may seem like cheating compared to the above, but Rudin is terse enough and skips enough steps that is actually still works. My priors indicate that a person’s memory is not so good that they can reproduce a fully rigorous proof of a theorem after seeing the real proof only once. But, it does sort of prime your brain so that you have at least seen what the main steps are, and can try to link each step together with meticulous detail.
Once you complete a chapter like this, if you have mostly understood what you have read and reproduced, you should be able to tackle some of the exercises. They will be too hard when you’re starting out, but a good metric for whether you are making progress is to acknowledge how many you can complete before you have to look something up. Personally, for me, I could solve no exercises from rudin until about chapter 3, at which point I could usually solve 1, or 2. Sometimes 3. By the time I got to chapter 7, I could usually solve the first 10 by myself if I worked pretty hard. By the time I got to chapter 11, I could do probably 3/4ths of the exercises in the chapter, with the larger constructive proofs still evading me. I would peg this at about a B-grade level of understanding.
Since you will likely need help on the exercises, try to approach them the way you approached the reading. Really do try to write them out without help, and spend a good amount of effort doing this. Learning math is about exposing yourself to the feeling of “Oh, NOW I see how that works” as much as possible. Simply copying answers won’t do this for you, but getting stuck for a good amount of time on a problem, then looking at an answer, or asking for help usually does. “Ah, so I was supposed to use theorem 6.17 here”.
The result of all of this work is that you won’t be scared of notation, even if its unfamiliar, and you will understand how a proof should look, when a step is valid or invalid. You might also learn how to think of your own proofs, but this is an ongoing skill. The point isn’t so much to be able to prove theorems, it’s to be able to understand them, and to understand how mathematics works as a formal system.
Oh, and do more exercises than assigned if at all possible. I didn’t for a long time, until I realized that my peers who were better than me generally did all the exercises. They didn’t work that hard on the supplementary exercises, but they at least did them and understood the intended solutions. This can be a nice “secret weapon” for exam preparation. Alternatively, solve exercises out of a different book, at a slightly lower level (Rudin is hard enough, to go harder is to go graduate level and graduate level means measure theory).
IV.
“Anyone who survives a year with Rudin is a mathematician”. Well, that is a nice badge of honor, but I am not sure I agree. Maybe a better statement is “Anyone who survives a year with Rudin will no longer be afraid of math, understand coherent thinking, and what it really means to prove something.” I would say that “mathematician” implies that someone can come up with original proofs to new problems, which Rudin probably won’t teach you how to do. That skill is ongoing. But, I agree that “Mathematician” certainly sounds better than a disjointed sentence of personal qualities.
At the end of my year with Rudin, I felt almost exactly this way. Math was no longer scary, just difficult. I could come up with proofs more quickly and easily than my peers who had not slaved away with me through a year of the blue book. Picking up a new mathematical skill came more quickly and easily, and statistics and probability were suddenly a lot more tractable, and not because I was suddenly a lot better at calculus.
Make no mistake, a year with Rudin is harrowing, for even if you try to cheat your way through it, there are always exams, and those do demand some understanding and originality (under time pressure, no less).
“THERE ARE BETTER BOOKS OUT THERE”.
Maybe. Abbott’s book is pretty good. I am not sure why people still recommend Apostle (too old school and wordy). But terse definition-theorem-proof mathematics never goes out of style. And Rudin does teach you a lot of things. It teaches you how to think, how to prove, how to work hard, how to read a dense book, and how to ask questions. It teaches you whether mathematics is really for you. It teaches you focus, and dedication. It even teaches you how to be alone, the yellow hue of your desk lamp shining on your scratch paper, illuminating your room.
Principles of Mathematical Analysis (Walter Rudin): 4.5/5
-0.5: too much personal suffering
Probability Measure 