# Difficulties with real analysis. Please give me an advice (Book recommendations)

Please excuse my terrible english skills.

About a year ago, I decided to study basic mathematics. At this time, I was not even able to find a solution of some tricky linear equation.

I studied basic algebra, euclidean and analytic geometry, linear and quadratic, polynomial functions, trigonometry and some basic calculus. (Also stochastics, but we dont have to regard it) These stuffs cover most of the high school mathematics at least in my country. And it should be sufficient knowledge to study so called introductory real analysis.

I never had college calculus. And I am also not willing to spend my time for it. (Stewart, Thomas.. etc.)

Now I am learning university mathematics for myself since few months due to interests in the subject. First I studied naive set theory with the first chapter of Munkres,Topology. Then I jumped directly into Baby rudin. I didn't have any serious problem until chapter 2, basic topology was pretty enjoyable. I could solve most of the exercises. Nevertheless I got stuck in Theorem 3.20 (Some special sequences). I can just barely understand things from this part until now, although I tried and tried.

Is there some easier book than Rudin, which is written for absolute beginners with weak calculus knowledge? And can someone explain me why I can't go through that chapter? I want to learn Real Analysis. Please help me

• A slightly easier (but still not easy) book than Rudin is Michael Spivak's Calculus (NOT his Calculus on Manifolds... that's an entirely different beast). THis is an extremely well written book, with few errors, and VERY VERY VERY good problems. Even though I finished the book 2 years ago, I still find myself referring to it from time to time :) – peek-a-boo Mar 18 at 23:14
• I think you are quite right not to go with those enormous, expensive, thick college Calculus textbooks. They are dreadful ! – Simon Mar 18 at 23:37
• I quite like Kenneth Ross's "Elementary Analysis: A Theory of Calculus." The book is very well-written, and much, much easier than Rudin, which I personally view as a book that is far better suited to a second course in analysis than a self-study. Spivak's Calculus, as peek-a-boo suggested, is another outstanding book. – John P. Mar 18 at 23:37

## 3 Answers

I really like "Understanding Analysis" by Stephen Abbott:

https://www.springer.com/gp/book/9781493927111

Some say it is not rigorous, but in my opinion it is perfectly rigorous enough. It also motivates the definitions and theorems very well, hence the word "Understanding" in the title.

I also recommend Tao's "Analysis I" and "Analysis II":

https://www.amazon.com/Analysis-Third-Texts-Readings-Mathematics/dp/9380250649

I also really like Carothers' "Real Analysis":

https://www.amazon.com/Real-Analysis-N-L-Carothers/dp/0521497566

All authors really convince the reader that a) They understand the topic in great depth, and b) They want to share their understanding with you. This is surprisingly rare !

• You were faster just by a few seconds :-) Definitely my favorite (not only) at the moment - I teach a full course of MA (four semesters) and I use this one for a part of it. I mean Abbott's - I don't like Tao's that much. – Roman Hric Mar 18 at 23:20
• Isn't it excellent ?! The real men at the beach will kick sand in our faces for going with the wimps' option instead of Rudin, but I tend to think they aren't very happy inside :( – Simon Mar 18 at 23:22
• Definitely... I never use just one source (some students might not be completely happy about that) but the one by Abbott is a true delight. – Roman Hric Mar 18 at 23:27
• Didn't he write Flatland? – Chris Custer Mar 19 at 2:31
• I think that was Edwin Abbott, much earlier. – Simon Mar 19 at 2:39

I was also going to suggest Abbott's "Real Analysis" book because it is intuitive and inviting for someone who is having their first exposure to real analysis. It also has some very nice exercises that are great for strengthening one's understanding of the key concepts (it asks to fill in details for proofs given in the book, which one would always do anyway in an ideal world).

There's "baby Rudin". It's a good book. Principles of Mathematical Analysis series Then when you get more advanced, as the name suggests, there's another book.

There are others, some I was aware of. I think Royden has one. When I started grad school at UCLA, we used Wheeden and Zygmund (or something like that). It had all the monotone, dominated, etc. convergence theorems. We learned about $$L_p,l_p$$, convolutions, kernels etc. etc.

Oh yeah, and don't forget Folland. He's up at U of Washington. His book is impressive.

Oh, and how could I forget, Rosenlicht. He was an excellent (as usual) Berkeley professor. Pardon my not having read what you wrote more carefully before posting. But I really like Rosenlicht as an intro to the subject. Besides it having sentimental value. It's a beautiful little book, published by Dover.