Good real analysis books recommendations please.

[Chsoviz0716]

Hello,

I'm not quite sure if this kind of question can be posted on this board. Please excuse me if not.

I started studying real analysis with Rudin's Principles of Mathematics which was relatively compact. Then I bought Apostol's book which was much more helpful because it was more thorough and detailed.
So I was wondering if there are any other real analysis books that are very thorough. I don't mind studying very thick books. If you have any recommendations as someone who studied this area before me, could you let me know some of the books you consider good? I would really appreciate it. Thank you.

 

[DivisionByZro]

Here's a blog entry from one of our mentors, micromass:

https://www.physicsforums.com/blog.php?b=3654

Hope that helps.

 

[micromass]

Here's a blog entry from one of our mentors, micromass:

https://www.physicsforums.com/blog.php?b=3654

Hope that helps.

Hmmm, that reminds me that I need to rework that blog. There are some good books I missed...

 

[qspeechc]

Pugh's Real Mathematical Analysis. It is at the same level as Rudin, covers more or less the same topics, but it is a much better book. It provides more intuition and insight, better problems, and especially the coverage of several variables and measure theory is much better, but the other topics are also better covered.

Best of all, it's much cheaper than Rudin.

 

[Jolb]

Rudin's PMA isn't just real analysis; the entire book is basically written in the context of complex numbers. I love Baby Rudin, because even though it's a very tough book (he doesn't spoon feed you at all) and even though the exercises are very challenging, it's basically impeccable (aside from the measure theory/lebesgue integral chapter) and has definitely stood the test of time. Apostol is okay too, but I don't like his treatment of Lebesgue measure and Lebesgue integration (he manages to only define "measure zero"; you never learn what measure actually is! and measure theory is important)

I would recommend Zygmund and Wheeden's Measure and Integral. The book goes through most of the important results of analysis and does a thorough treatment of Lebesgue measure and integral.

 

[Chsoviz0716]

Thank you everybody who replied. I looked up the books mentioned above and was a bit puzzled.
Some books like Rudin's PMA or Apostol's cover the basic materials such as sequences or differentiation or Rieman-Stieltjes integration, while some other books I found cover very little of those, but cover measure theory or Lebesgue integration in more detail.
Should I conclude that there are two types of real analysis books; easier ones(the ones I mentioned first) and relatively hard ones(the latter)? Or is it going too far to distinguish those books into different categories?

 

[micromass]

Thank you everybody who replied. I looked up the books mentioned above and was a bit puzzled.
Some books like Rudin's PMA or Apostol's cover the basic materials such as sequences or differentiation or Rieman-Stieltjes integration, while some other books I found cover very little of those, but cover measure theory or Lebesgue integration in more detail.
Should I conclude that there are two types of real analysis books; easier ones(the ones I mentioned first) and relatively hard ones(the latter)? Or is it going too far to distinguish those books into different categories?

Uuuh, you'd be the first one who would call Rudin an easier book...

But yes, there are differences between lots of real analysis books. You will most likely have to read multiple books on the subject.
In any case, I don't think it's worth it to waste much time on Riemann integration. Riemann integration is such a stupid and flawed method compared to the other things we have. The faster you cover Lebesgue, the better in my opinion.

 

[Chsoviz0716]

Uuuh, you'd be the first one who would call Rudin an easier book...

But yes, there are differences between lots of real analysis books. You will most likely have to read multiple books on the subject.
In any case, I don't think it's worth it to waste much time on Riemann integration. Riemann integration is such a stupid and flawed method compared to the other things we have. The faster you cover Lebesgue, the better in my opinion.

oh, I meant the 'material' it covers is relatively easy, not how it presents the material. May be this isn't true either, it's just the impression I got from my short time of studying.
Anyway thank you very much for your advice. I guess I will have to just study more to better see the bigger picture of analysis.

 

[qspeechc]

Measure theory and Lebesgue integration come after a book like Rudin, and Riemann integration. Riemann integration is one of the things you must know.

But look, I read your post again. You have Apostol and Rudin. Start with Apostol (do you have Apostol's calculus or analysis book? Either wy start with Apostol), then Rudin, then some book that covers measure theory and Lebesgue integration (Apostol's analysis book covers them, I think). Then you will know everything you need to know in undergrad real analysis backwards. Don't waste your money buying more books, I think the two books you have will be more than enough.

 

[Chsoviz0716]

Measure theory and Lebesgue integration come after a book like Rudin, and Riemann integration. Riemann integration is one of the things you must know.

But look, I read your post again. You have Apostol and Rudin. Start with Apostol (do you have Apostol's calculus or analysis book? Either wy start with Apostol), then Rudin, then some book that covers measure theory and Lebesgue integration (Apostol's analysis book covers them, I think). Then you will know everything you need to know in undergrad real analysis backwards. Don't waste your money buying more books, I think the two books you have will be more than enough.

Thank you for your advice. I have Apostol's analysis book. And yes it seems Apostol's book covers measure theory and Lebesgue integration, but I heard it's somewhat limited in its content compared to other books that are mostly devoted to those subjects(measure theory and Lebesgue integration).

 

[mathwonk]

Real analysis is a big subject. I like Dieudonne's book Foundations of modern analysis, even if it does not cover lebesgue integration. It covers a lot of useful material. There is also a second volume that covers lebesgue integration.