# How to best learn analysis? [closed]

I am currently studying Analysis through Pete L Clarks's thorough notes, which he bases on Spivaks Calculus. I only took Calculus and linear algebra and am going to be taking a Topology course at my uni. I want to learn the material covered in Spivak and then Algebra I and II in order to have the prerequisite knowledge and proof writing skill set to enter the Topology course.

Pete's notes are great but I am wondering how I can best cover the material as my approach to studying Calculus and linear algebra doesn't seem to be effective with this material. For example, in proving Euclids Lemma using the well ordering principle, what I am currently doing is trying to understand the basic strategy used and then to follow each and every step of the proof without moving forward until I fully understand why a step was taken.

This has been working OK so far but I'm wondering if there's anything different I should make sure to be doing, such as rewriting the proof after several times; sort like how we learned the quadratic formula in high school.

I do long distance running at a high level and the training method is to: 1) build endurance; 2) build speed; and 3) work on form. There are specific workouts which you can do to build the area you want to develop. Do those things consistently and you'll see significant improvement.

Is there a similar methodology to mastering advanced mathematics? I'm not looking to make the content easier, I quite enjoy the rigor of the material, but just a clear goal for what I should be accomplishing in each chapter.

## closed as off-topic by Jack, астон вілла олоф мэллбэрг, JonMark Perry, Claude Leibovici, user91500Mar 14 '17 at 10:04

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• "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – Jack, астон вілла олоф мэллбэрг, JonMark Perry, Claude Leibovici, user91500
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I am happy that you like the notes and more so that you're inspired to study this material. I have to admit though that I don't have any ready-to-offer self-study strategy for these notes. As you know, they began as supplementary lecture notes for a course I taught out of Spivak's Calculus text, and then they've been expanded upon in various ways since the course ended. I guess I want to be honest and say that for the typical student who would take such a course -- i.e., a bright undergraduate who is looking to get into real analysis from a quite modest starting point -- reading the notes alone would be a poor substitute for taking the class.

(I may not be admitting that much. For instance, compare to Rudin's Principles of Mathematical Analysis: this is a very popular text, but it is difficult to study on your own. There is a reason we have undergraduate math classes, after all.)

It is easier for me to answer in terms of the way the notes are written and what is not in them, so I'll try that.

1) The notes were written as a supplement to another text that is sparing in its actual text and has copious exercises, some of which are very challenging. So the notes go the other way: a lot more is proved in the text itself, and there are not nearly as many exercises. The advantage of that is that if what you want to see are more worked out examples of nontrivial arguments involving $\epsilon$, $\delta$, sups, Riemann sums, and so forth, there they are. The disadvantage is that there aren't nearly as many exercises as in most texts. Moreover the exercises appear intertextually only and tend to come up as direct responses to what was just done in the text rather than general reinforcement. Looking back at the text now, I find that there is exactly one exercise in Chapter 4 (Continuity and Limits). Uh oh! You'll almost certainly need to supplement with another text (e.g. Spivak!) here.

2) Spivak's text is wonderful at keeping things interesting. As mentioned above, he can keep things interesting with lively problems even when there is little to nothing going on in the text itself (e.g. Chapter 3 on functions). But once the text gets properly underway there are many wonderful digressions: for instance, he has an entire chapter each devoted to planetary motion, the irrationality of $\pi$ and the transcendence of $e$. Rudin's Principles, written in the 1950's, pioneered a much more streamlined approach. It gets plenty exciting eventually: Chapters 7 and 8 are set in Mount Orodruin, an amazingly gripping climax to all that has come before. Most analysis texts since Rudin have decided to either cover the same content as the first eight chapters of Rudin in twice the space or seem eager to get on to the graduate-level measure and function theory. The breadth and depth of undergraduate level real analysis that one finds in e.g. the Calculus and Analysis texts of Courant and John (several thousand pages worth of material!) is mostly absent from modern texts.

Like Spivak, I wanted my notes to reflect the breadth and depth of undergraduate level real analysis. So it has a lot of excursions as well, but mostly different ones from Spivak. In particular I wanted to give a careful, lively treatment of certain topics that by now seem never to get a theoretical treatment in the undergraduate pure math curriculum: e.g. partial fractions, convexity, Newton's methods. So for me the most important chapters of the text are Differential Miscellany, Integral Miscellany and Serial Miscellany: i.e., three chapters full of (mostly) logically unnecessary diversions on derivatives, integrals and infinite series.

What does this mean for the self studier? I think it means: reading the notes carefully from start to finish is probably not a good idea, because things that are very ancillary are discussed along with things that are very important. As one example, I give a full blown treatment of the partial fractions decomposition, developing the portion of factorization theory in a univariate polynomial ring over a field necessary to do it, quite early on (Chapter 3). I put that in because in freshman calculus one uses the partial fractions decomposition to integrate rational functions whereas no calculus text I know of written in the last 50 years proves the partial fractions decomposition. Okay, but where is this used later in the text? Well, it would be used in Section 9.3.2 (integration of rational functions)...except that that section is currently blank. (Once you know the PFD, the discussion of integration of rational functions is exactly as in freshman calculus, except that giving a general treatment rather than what turns out to be a comprehensive-enough recipe necessitates a bunch of horrible notation, as I did cover a bit in my course. Also you have to pretend that you know how to factor polynomials over $\mathbb{R}$...which you actually don't without further theory e.g. due to Sturm.)

So you should not read carefully from cover to cover, but rather spend the most time on the material that is the most important. I think the text does a pretty good job at making the most important material easy to find: on a first reading I would concentrate on Chapters 4 (Limits and Continuity), 5 (Differentiation), 6 (Completeness), 8 (Integration), 10 (Sequences) and 11 (Series).

I hope that is at least somewhat helpful. Good luck.

• This is very helpful!! Thanks again – john fowles Mar 14 '17 at 3:16