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I'm self-teaching math at home. After learning Calculus by MIT OCW and a bit of easy Linear Algebra and set theory, I have decided to study Analysis.

As many have recommended Baby Rudin, I've tried to study Analysis with it. But it was so tough and hard that I could only solve 1 or 2 problems on the exerices. I thought that this is not the book which I can deal with. So I look around another books.

Here's a list: Tao's Lecture Notes(Solution available and it seems to be approachable) #Since I'm studying only by myself, so Solution availabiltiy is somewhat important as I should check my answers and be given some feedback. Marsden(I didn't carefully read the contents, so I don't know what the level of this book is. But I've heard after some chapter , it has lots of errors.) Abbott(it looks like the easiest one I've seen among Analysis books) Apostol(It deals with a lot of stuff, which I'd like to know. But the exercises looks difficult for me.)

Which book should I use? If I choose a book and after studying the book, should I study harder Analysis book like Rudin?

Thank you.

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    See http://math.stackexchange.com/a/2814/589. – 2017-02-18

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What it seems you're looking for is a book on analysis intended for students who know calculus but have little experience doing rigorous math. A good book that is often used for this purpose in North American universities is Elementary Analysis by Ross.

An alternative approach would be to use a book that is expressly intended as a transition to rigorous math but which doesn't focus on analysis, and then use a more difficult analysis book (like Apostol), which you might be able to handle after further preparation, or one of intermediate difficulty like Lang's Undergraduate Analysis. I think one of the best books of this kind is Journey into Mathematics by Rotman. It has a complete solution manual.

Edit: I'll add that Spivak's Calculus can reasonably be used as an introduction to basic analysis, and it has a solution manual.

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Undergraduate Analysis - Lang

Postmodern Analysis - Jost

A Course in Mathematical Analysis (3 volumes) - Garling

Analysis (3 volumes) - Amann & Escher

Make sure that you handle well calculus and linear algebra before entering into more abstract stuff.

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    My reccomendation was Jürgen Jost, *Postmodern Analysis*, Springer (2006), and I now see that you write it before than I...+1. I didn't read this book but I've read in the past, Jost, *Partial Differential Equations*. Then I want to recomment it because the book on pde was very good. Good luck to you and @Grimza – 2017-02-20
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Choosing the right book is a very personalized experience. I think it is best for you to pick the book which you can follow comfortably and has sufficient motivation and examples. Once you are comfortable with the material, you could try solving problems from some other book which you think is harder.

Maybe it is just me, but I do not find Rudin very comfortable. It seems dry and provides no motivation for the material. I do not think a textbook should be presented as a bunch of theorems and proofs.

The most intriguing book I found was Understanding Anlaysis by Stephen Abott. I have also seen it being recommended elsewhere on StackExchange.

There is a rather soft book on Analysis called How to think about Analysis. It is not a textbook, per se, but if you feel overwhelmed, you could look into it. It discusses some basic things about Analysis.

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I like Apostol for its exposition, and especially for its exercises; I know that my math department likes to take some of the questions from there for the qualifier. Abbot overall seems like a more appropriate choice for an "approachable" first analysis text, since its exercises and choice of material is easy to digest the first time around.

I think that Carothers is also a good choice (even if you only do the chapters up through uniform convergence); perhaps as a "second book", before either Apostol or Rudin. First of all, it's the one that I first learned from, so I like it. Second, it tends to emphasize the (more abstract) metric space/topology framework a lot more than Abbot (for example). You might find that you're interested in the book's treatment of Riemann-Stieltjes integration and BV functions, but both of those can definitely be skipped if you want to stick to the essentials.

It really helps to have more abstract, "fuzzier" view of things sometimes, even if you only pick that up the second time around. In particular, once you understand the idea of an "arbitrary metric space", you'll notice that a lot of proofs run through the same ideas, slightly altered to fit the situation. With topology, you might start to see "past the epsilons and deltas". Also, it's vital for functional analysis (and PDEs) that you have a feeling for metric spaces that aren't $\Bbb R^n$.