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

Question

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

Answer 1

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 !

Answer 2

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.

Answer 3

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).

Answer 4

user 403337's answer appears unfitting, because OP already wrote that (s)he found Baby Rudin and Rudin too difficult. I disagree with the recommendations of Folland, Baby Rudin, Royden. These are for a second course in Real Analysis, not a beginner.

Stephen Abbott's Understanding Analysis has a solutions manual, but FOR INSTRUCTORS ONLY. You can find it online, but Abbott did not intend it for anyone to download. I dislike how it's wholly black and white.

I love Amol Sasane's The How and Why of One Variable Calculus, particularly its diagrams in color. This title is misnomer — This book covers Real Analysis, not merely one variable calculus.

The exercises are plentiful, well-selected and well-constructed. Detailed solutions to all exercises are provided; they fill the last 150 pages of the book.

Next comes more than one hundred and fifty pages assigned to providing the full solutions to every exercise given in the previous six chapters.