Real Analysis Textbooks

[BenedictProsper]

I am looking for a real analysis textbook. Any recommendations?

[ZomgRogged]

Principles of Mathematical Analysis by Walter Rudin seems to be what people here consider standard.

[Mankins]

In my analysis class we used Introduction to Analysis by Maxwell Rosenlicht, and Rudin was an optional text used mostly as a reference. Rosenlicht is much cheaper, but Rudin is considered the standard.

[jamielimzh]

I personally like the "Introduction to Real Analysis" by Robert Bartle and Donald Sherbert. Think it's probably the most readable text if u havent been taught any form of real analysis formally before. The downside of the text is that it doesnt cover much on topology.

Btw, I'm quite surprised nobody in this forum seems to have mentioned this as a text for real analysis... cos from what I know, this is a commonly used text for the first course on real analysis by most math departments

[ZomgRogged]

Originally Posted by jamielimzh

I personally like the "Introduction to Real Analysis" by Robert Bartle and Donald Sherbert. Think it's probably the most readable text if u havent been taught any form of real analysis formally before. The downside of the text is that it doesnt cover much on topology.

Btw, I'm quite surprised nobody in this forum seems to have mentioned this as a text for real analysis... cos from what I know, this is a commonly used text for the first course on real analysis by most math departments

My school uses this for Elementary Real Analysis I believe...

[philecon]

You may want to look at Understanding Analysis by Stephen Abbott.

[gramophone]

In my school, "Real Analysis" is a course for measure theory, Lesbesgue integration and some preliminary functional analysis results. For this course, Royden is a standard text. The textbook wriiten by Stein (with another coauthor...I forget his name) is also a good start.

If "Real Analysis" means a course for introduction to metric space, definition of differentiation and Riemann integration and so on, I think every textbook mentioned above are ok, but I prefer Rudin's PMA. Marsden's textbook is also good.

[breakz]

Tossing in a vote for Rudin. Also nominating "Mathematical Analysis" by Apostol.

[greeneggs]

You're going to need to go through Rudin, but Rudin isn't a very good book to learn/start from. The proofs are elegant, but unmotivated and everything is provided without intuition. I would suggest starting with "Analysis: With an Introduction to Proof" by Steven R. Lay; it covers mostly the same topics as the first semester of Rudin does, but much easier - so it's a great thing to possibly carry around with Rudin or read just before. Really no need to buy it, because you'll never need to reference it - just get it through a library read it once and you'll understand it; otherwise an older edition is fine too.

Another of my favorites is "A Radical Approach to Real Analysis" by David Bressoud. It's really a great book, it tackles Real Analysis in a completely different way, basically taking you through it chronologically as it was discovered/developed; and providing you with the history and intuition - as well as mistakes great mathematicians made - necessary for you to really appreciate what you're doing and why you're doing it that way, and what other ways you could have possibly done it. That's something I think is lacking in pretty much all other books, especially definition-theorem books like Rudin. I would also recommend Bressoud's other book "A Radical Approach to Lebesgue's Theory of Integration" for some historical background on measure theory and more advanced real analysis topics for after you finish with Rudin.

If you're basically looking for a cheap version of Rudin you can use "Introduction to Analysis" by Maxwell Rosenlicht or "Introductory Real Analysis" by Kolmogorov and Fomin; both of which you can get on Amazon for ~$10. In all honesty they are just as good as Rudin and cover pretty much the same topics in the same order; just for whatever reason they aren't used very frequently, so Rudin gets to charge $140 for his. If I was a teacher and cared about my students' pockets I would use one of those books. Otherwise you can always find a scanned version of Rudin online if you're any good at finding that kind of thing; not as nice as having the book in your hand, but it would be free. One good thing about Rudin is you can find a lot of solutions and companion notes for it; be careful though - there's a lot of incorrect solutions on the web too, so don't just blindly copy.

One last book I would recommend is "Counterexamples in Analysis" by Bernard Gelbaum and John Olmsted. It's a really nice book for seeing some classical counterexamples and can save you a lot of pain if you're trying to figure one out or wanting to prove something that actually isn't true.

There's also the possibility of "Real Analysis with Economics Applications" by Efe A. Ok. It got good reviews on Amazon, but I don't know anybody who's looked through it so I have no idea if it's any good.

[breakz]

I figured Rudin was just the upper division hazing mechanism used by math depts, so why question it? :P