Introductory Analysis textbook

I know you said you distrust amazon reviews, but after reading the first one there, by Ian Smythe, I wonder why. It seems hard to imagine a better, more useful review than that. If you haven't done so, I recommend you read it.

 

I don't know if this helps, but here is a review from the MAA:
http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=66436

 

I do not know the person but I still think I can gain information from reading it. There are two kinds of statements one can make when evaluating something. You can say "boy this is the greatest book in the world!" and such a statement has essentially no value unless you trust and understand thoroughly who is saying it and what they mean by it. The second type of statement is descriptive, e.g. (I am making this up) "this book devotes one preliminary chapter to developing the real numbers from the point of view of dedekind cuts. then it goes on to discuss the concept of a general metric space in the next 3. There are numerous examples, such as R^n, and L^2 spaces of square integrable functions. Each section ends with illustrative problems for the material in the chapter as well as more challenging ones to test both ones grasp of the topic and ones creativity. Most of our senior level undergraduate class apparently found the book about the right level after background in one and several variable calculus."


See what I mean? even if you don't know the person there, you can pretty well learn something. I recommend you practice reading things by people you do not know, to learn how to derive information from them. If you just look for "reliable" sources, and take whatever they say as true, you will be less well off in my opinion.

Someday you may have a job as a faculty member evaluating hundreds of applications for scholarships and jobs, and you cannot possibly know all of the letter writers. In fact in my experience those people who just ask their friends what to think about someone do less well, because of possible bias, than those who try to understand the relatively anonymous references.

to be brief, you can usually trust someone who says one book is longer than another, or more detailed, or has more examples, but not someone who says one book is better than another.

Or look at the book yourself.


after reading the maa review linked above i still recommend you read some of the 34 amazon reviews. such as the one by randy ringstad. there you will get a student's perspective in addition to the professorial perspective in the maa review. You should get all the information you can. but as i said before you should essentially ignore reviews which mainly say the book is great, or is horrible. those are just raves or vents, not reviews.

 

My so far short experience with analysis books is that just one book won't answer all your questions. In fact, no book will answer all of your questions or explain every little detail for you. What's going to happen is that all books will leave some gaps for you to fill in. Some of them will say it, e.g. "we will leave this as an exercise" or little notes like "verify!". Some of them just won't say anything, but for your own understanding, you should try and figure out why one thing implies another. Then it comes out to how much stuff do they leave out, and how much time you have. A book like Rudin's leaves a lot of stuff out, so obviously it'll take you more time to figure out his proofs. This can be a good exercise, but it can also be wasteful. Wasteful because there are some things you need to be shown by someone, and there's no merit to discovering them by yourself (though some people would argue about that). Anyway, it seems like the textbook you linked is what's your professor is going to use, so there's really no way out of it, whether it's a good book or not. However, if you take this class with a professor who's helpful, then sometimes it doesn't really matter how good the book is.
One last thing though, I found it helpful to have other books at my disposal. It helps a lot when one book is such that there's a discussion of every topic. Some introduction to what this topic is all about, etc. https://www.amazon.com/gp/product/0387950605/ref=oh_o04_s00_i00_details" has that kind of discussions in it.
Anyway, what mathwonk wrote about reviews is true and very helpful.

 

I haven't used Bartle's "Introduction" but I have used his "Elements" text (which is a little higher-level, I believe). I think he writes very well so I would hope that it is true for the other book as well.

Edit: I just noticed eliya's link to Abbott's book. It is a very good introductory text. I would also recommend it if you haven't been exposed to much analysis. It is a perfect entry point for someone who has a solid background in Calculus but hasn't had a more theoretical "Spivak" type course.

 

While I like Abbott's discussions, I don't know if I'd recommend it. Some of his definitions are actually shown in other books to be theorems. I don't think it's necessarily a bad thing, I just found it confusing. Another book that people like to recommend is https://www.amazon.com/Elementary-A...111/ref=sr_1_1?ie=UTF8&qid=1321286111&sr=8-1", but I never read it. Go to your school's library, look at the calculus/analysis section, pick a few books, and see which one has the kind of style that suits you best. You only need one book for definitions and theorems, the rest are just books you can have a discussion with. Of course, you can't talk to a book, but you get the idea. Good luck.