# How can I pick up analysis quickly?

I have a 2-3 week recess from university for winter break. In this time, I would like to learn analysis, starting with Walter Rudin's Principles of Mathematical Analysis, and then, if at all possible, continuing with Walter Rudin's Real and Complex Analysis. If necessary, I would be willing to complete the second book after returning to college (that is, outside of the 2-3 week time frame).

A few questions come to mind:

How reasonable are these goals?

My background in maths is an elementary Moore method single-variable calculus course, and the beginning of (undergraduate) introductory real analysis. However, most of my time during the break will be available for mathematics. Is only the goal of completing the first book reasonable, with the second book requiring additional time?

Is Principles of Mathematical Analysis sufficient for reading Real and Complex Analysis?

If not, what else should I know?

What advice can you give me?

I'm reading these primarily for entertainment, and I hope with this to learn enough mathematics to do interesting things. (I am a maths student in college, but have just started undergraduate analysis. My courses do not use either Rudin book.) This does not need to be advice on the books themselves, perhaps it could be advice on how to learn math quickly (and properly) if one has sufficient time to think about it exclusively.

• I recommend you to read Folland's real analysis book at the same time. If you think you lack understanding in topology, you can read Munkre's Topology, which doesn't take long to read. For complex analysis part, I recommend you to read Ahlfor's or Stein's book before. – Math.StackExchange Dec 14 '14 at 5:55
• In short: 1. This is not reasonable. 2. Principles is terse and dense; use it as a reference, but not for self-study. 3. Why do you want to read Real and Complex Analysis? 4. I'd recommend something by Spivak. 5. The book recommendations by others are good too. – aes Dec 14 '14 at 6:20
• 1. I don't think you can expect to pick up analysis quickly, unless maybe you are Ramanujan. 2. The question of using Rudin for self study has come up on this site many times, and the consensus is that while that book has its good points, it is not a good choice for self-study. – MJD Dec 14 '14 at 7:35

Principles is an excellent text, but I don't think it's well-suited to self-study. There's nothing wrong with it, honestly, and you'd probably be fine reading it, but to me it's one of those many excellent texts that doesn't really "stick" to the reader that well. It's a better text for an intensive undergrad course with a good professor.

There's a definite distinction between a good text for a lecture course and a good text for self-study. Usually the former contains a broader list of topics and excellent problems, but is very terse and emphasizes logical structure and organization over a readable narrative. It's meant to be studied from, to clarify the structure of the subject to the student and help hammer the points home. Rudin's books fall into this category. For actually reading the material and getting the most benefit from it, I prefer books that take a more classical approach. They tend to have more motivation, examples, and a clear narrative from the author. For this I would recommend Pugh's Real Mathematical Analysis. After this, if you're interested in learning more analysis, I'd recommend Royden. Folland is a great text as well, but falls more under the first category (great reference, good for a lecture test, but I found it difficult to read on my own).

A couple of other alternatives:

• Consider reading Stein's analysis series. It's aimed at an undergraduate level, and would be a better place to start than going straight into Rudin if you don't have a good background in analysis. Once you've seen enough analysis, it's not too difficult, but the first encounter can be rather discouraging. (I was a terrible student in my first hard analysis course. I was lazy and didn't put the work in. Expect your first run-in with analysis to be extremely frustrating, but don't get discouraged. With enough work, one day it all "clicks".)

• Prof. Su from Harvey Mudd has a first semester analysis course up on YouTube. Harvey Mudd has one of the best undergrad math programs in the world, and it's evidenced by his lucid teaching style. He somehow makes even Rudin easy for students seeing it for the first time.

• You could edit that into your answer. Comments aren't the best place for expanding your answer. (Given your relatively low reputation, I thought you might not know yet.) – 11684 Dec 14 '14 at 9:17

Rudin's Principles of Mathematical Analysis is an extraordinarily onerous book. Even the parts that are not hard require a lot of work.

Also, it does not cover complex analysis at all. Even in the parts where it deals with complex numbers. For example if you study the complex-valued function $x\mapsto e^{inx}$, where $x$ is a real variable, you're doing real analysis, not complex analysis. Complex analysis is a subject with a surprisingly different flavor from that of real analysis.

• This doesn't quite address the Alexandre's question about the feasibility of learning the material quickly; on the other hand, that's largely a matter of opinion so perhaps best left out. – David Richerby Dec 14 '14 at 13:56