# Real Analysis

1. Jun 25, 2008

### qspeechc

Hi everyone.
I'm taking my first course in real analysis soon. I have a bit of linear algebra and advanced calculus under my belt. I would like to get a book as a companion to the course. What would you recommend? I've heard wonderful things about Rudin's Mathematical Analysis, but is it suitable for a first course in real anlysis (no complex analysis)?
EDIT: the course covers:
Infinite sequences; functions, limits and continuity; the derivative; infinite series.
Thanks.

Last edited: Jun 25, 2008
2. Jun 25, 2008

### Werg22

Do you know if the theory of limits and continuity is general (over any metric space) or simply concerned with R?

3. Jun 25, 2008

### dx

I recommend Michael Spivak's "Calculus". Rudin is not suitable for a first course, nor is it very insightful compared to Spivak (in my opinion).

4. Jun 25, 2008

### ehrenfest

Yes Rudin's "Principles of Mathematical Analysis" is a standard text used in undergraduate real analysis and I would highly recommend it. You will however meet a lot of complex numbers in that book and perhaps that is why you were concerned. Basically Rudin proves many of the results in that book in general for the complex numbers system but he doesn't really use any of the properties specific to the complex field the you would use in complex analysis. Most of the book is okay if you want to just work in the real system.

5. Jun 27, 2008

### Pseudo Statistic

Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger is a pretty good book to read while doing a first real analysis course (and nice for an introduction to later stuff, like metric spaces).
Our set text was Spivak, which is a good book as well, though it isn't quite as concise as J&P.

6. Jun 27, 2008

### BryanP

For honors level real analysis, Rudin's text happens to be used frequently at my school.

For elementary real analysis, we use:

Required Text: Kenneth A. Ross, Elementary Analysis: The Theory of Calculus
Syllabus: In this course we will study basic notations of real analysis, limits and convergence, series, continuous functions, power series, differentiation and integration, and metric spaces.

It's not a bad book, you should check it out.

7. Jun 28, 2008

### arshavin

I like Real Analysis and foundations, by Krantz

8. Jul 6, 2008

### MCVal

Another vote for Rudin's "Principles of Mathematical Analysis". I used that text for my course in real analysis last semester, and really enjoyed the format; unfortunately for some, Rudin is fairly light on examples, but most of the end-of-chapter exercises are very doable, even for someone without a good background in writing proofs.

9. Jul 6, 2008

### symbolipoint

Tell us more details about the format of the Rudin book. Are there any siginificant examples during each chapter section? Does the format conform closely to a typical undergraduate/lower division Math book? Is the book reasonably thorough? Is there an answer key at the end of the book? Is there a solution manual? Does the book use derivations and proofs as one would typically find in lower division Math textbooks?

10. Jul 7, 2008

### qspeechc

Thanks for all the suggestions everyone. You've been very helpful.

11. Jul 9, 2008

### zhentil

If you want to learn basic analysis, Rudin's book can't be beaten. However, his style sounds like you would hate it. There are basically no examples, the proofs (while elegant) are concise to the point of not yielding insight to the newcomer.

If you're used to the standard undergrad format, which basically holds your hand (i.e. lots of examples, the problems are very similar to the examples), and that's what you're expecting from Rudin, you'll be in way over your head. However, if you can stomach the fact that you'll be learning by doing the problems (some of which are moderately easy, some are positively wicked if you've not seen the material before), you will not find a better book.

12. Jul 9, 2008

### zhentil

Also, if you want an answer key/solution manual to a real analysis book, you should not be taking real analysis.

13. Jul 9, 2008

### ehrenfest

I disagree. Firstly, you might get stuck on a problem. Secondly, everyone makes mistakes at times and if you don't have a solutions manual and no one is reviewing your proofs then these just go unnoticed which is not good.

You can find solutions to selected problems by searching for Real Analysis courses on the internet. MITOpenCourseware has one with solutions.

14. Jul 9, 2008

### MCVal

No, there are not, some relatively major theorems are proved, but as the guy after me stated, they tend to be really concise; there is nothing lacking in them, they just take the quickest path.

Again, no. Rudin quite frankly is very condense, covering from basic properties of real numbers up through Riemann-Stieljes integral related material in maybe 130-140 pages at most.

I would say so, but a lot of the nice/interesting theorems are in the end of chapter exercises.

No, and no. You can find solutions guides that people have written up( I know a few people who used them); however, in my opinion, they are not needed. This is because:

1) If you are taking a class that uses the book, and you cannot prove a theorem, you should ask your instructor to go over it with you, or possibly ask a student who has done it. Using a solutions guide WILL cheapen your experience on homeworks, as you will not be spending the time "getting used to" how one can approach theorems in analysis.

2) If you are self-studying, you should be confident of your own abilities to prove the problems and evaluate your proofs for correctness(Oh the interesting misconceptions one can cement if they don't do this).

15. Jul 11, 2008

### ehrenfest

In order to self-study, you need some level of mathematical maturity if you want to get much out of it but the fact that you want to self-study usually means that criterion is met IMO. Even if you aren't completely confident evaluating your own proofs, I wouldn't let that stop you from self-studying real analysis. That is (very roughly I know) like telling an artist not to explore a new genre of art because they aren't confident giving a "good" or "correct" critique of that art form.

Being a seasoned proof-writer or proof-checker is not a prerequisite for self-study. In fact self-studying is a good way to get better at that. Really all you need is interest and knowledge of calculus to dive into Rudin. You may want to later come back after you have have become a more mature mathematician but there is no reason to wait until your mathematical abilities improve beyond a certain level to start self-studying. Basically just being exposed to Rudin's rigorous expository style and being familiar with the theorems and understanding ideas from the proofs is good for a developing mathematician.

If there is a solution manual it can save you a lot of time and frustration regardless of how good your proof-writing abilities are.

16. Jul 12, 2008

### qspeechc

Thanks for the interesting discussion! As real analysis is only my second proof-based course I'm not sure Rudin is the way to go. Anyway, I was planning on using a book over-and-above the course notes we're going to be given.
Rudin's Mathematical Analysis sounds like the real analysis equivalent of Griffith's Electrodynamics. A book I found thoroughly frustrating.

From my own experience, no solutions at all are a real bummer. Especially for the novice like myself. Sometimes you are stuck on a proof for ages. You try every angle, you ask for help, but nothing works. Then you look at solution and the light breaks through your despair. Usually- I've found- solutions are very bare-bones anyway, and you have to fill in the details and make sure you understand it yourself anyway. As a beginner in anything, I think solutions- not necessarily answers- can be very helpful indeed.

I think Rudin may complement my course notes, but I'll have to see.

17. Nov 17, 2011

Actually, I would think that Rudin's book has the reputation similar to that of Jackson's Classical Electrodynamics over Griffith's! :P

18. Nov 18, 2011

### deluks917

19. Nov 21, 2011

### dextercioby

Nobody mentioned Berberian's book, which is not introductory, but very thoroughly written.

20. Nov 21, 2011

### qspeechc

Hello everyone. If you bothered to look, you'll see this thread is 3 years old. It's considered bad manners to bring up an old thread.

I did real analysis ages ago. And I know about Pugh and have been recommending it for many years.