What Real Analysis book do you suggest?

[AdrianZ]

Hi all,
I've been self-studying Rudin's Mathematical Analysis recently and have studied the first 4 chapters so far and I'm fine with the way it has developed the theory but the book lacks solved exercises and examples to be called a perfect book for self-studying. I have learned the general concept and the theorems it proves but I'm still not able to think in a way that a mathematician should think. I mean my skills are limited when it comes to solving analysis questions. I'm looking for a textbook that explains Real Analysis in a way that is suitable for studying analysis on my own and covers the same ideas in a self-contained way.

Thanks in advance

 

[micromass]

I'm quite a fan of "principles of real analysis" by Aliprantis and Burkinshaw.
It's quite different from Rudin, as it covers more measure theory, but it doesn't cover multivariable stuff that well. Still it has a very interesting choice of topics:

It starts with the real number system and it covers a bit of metric spaces. I like this, because I feel that every real analysis book must do some metric space theory.
Then it goes on to generalize all of this stuff to topological spaces. We also prove the very useful Stone-Weierstrass theorem.
The next two chapters cover measure theory and integrals. This might be difficult for a first time read, but I like the treatment.
We end with a bit of functional analysis: $L_p$-spaces and Hilbert spaces. I'd say it's enough to wet your appetite in the topic.
And finally there are some nice extras in the final chapter Like Radon-Nikodym and Riesz representation.

Obviously, one book is never enough to know real analysis. You'll need to do some extra reading on topology, functional analysis, measure theory, etc. afterwards. But I think this is a nice book to start your research on these topics.

The exercises are do-able. I mean with that that I had a quick look at the exercises and I immediately knew how to solve most of them. This contrary to Rudin, where I always have to think a little while before I find the solution (if at all).

 

[Gib Z]

A good place to start for a true beginner is Apostol's "Mathematical Analysis". If you feel comfortable with much of that content or are up for a challenge, you could begin straight away with Royden's Real analysis textbook. It covers all the basics (sequences, series, uniform convergence and the like), and it also covers Metric space theory (which sets one up for a future course in topology quite nicely) and it covers basic Measure theory as well.