Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm self-studying Rudin's Real and Complex Analysis. I've finished the first two chapters so far, and I don't have any major problems understanding the definitions and theorems. I can prove the theorems on my own. Exercises in chapter 1 were OK, but I'm finding the exercises in chapter 2 to be very difficult. To give you an example, one exercise expects you to come up with the generalized Cantor set on your own. Another is a proof that was published in a journal.

Are such exercises the best way to learn for a beginner? Or is it better to start with a simpler set of exercises that test your understanding of the material before you venture into more difficult things? Should I augment my study with another book that has easier exercises?

I'm feeling frustrated and would like some guidance here. Thank you.

share|cite|improve this question
While the book is about real analysis and measure theory, the question itself has nothing to do with either. It's a good [soft-]question though. – Asaf Karagila Sep 25 '12 at 11:26
Difficult exercises are often not a good start in my opinion. You should also find on a source of easy exercises too. That would enable you to just test your understanding of the subject you're learning before moving on to difficult exercises. Or ideally, find a source of exercises of progressive difficulty. – Joel Cohen Sep 25 '12 at 11:45
up vote 30 down vote accepted

I think difficult exercises are essentially bad in general: they tend to discourage the student rather than check the understanding of the material.

It so happens that I too tried to read that book by Rudin in my fourth university year.
I found the reading very hard going and there were many exercises I couldn't do: this affected my morale very negatively.
Retrospectively, I find this book dreadful pedagogically and the worst offence is that there are no pictures : this is a mortal sin in a book on a geometric subject like complex analysis.
(In fairness I should add that I do use it as a reference now: it contains sophisticated beautiful results like the theorems of Müntz-Szasz and Mergelyan, which are not often proved in books on holomorphic functions.)

On a more constructive note, let me mention two great books you might consult:
$\bullet $ Lang's Real and Functional Analysis which contains an astonishing wealth in material (including the Haar integral and Schwartz's distributions)
$\bullet \bullet $ Remmert's Theory of Complex Functions , written by a genuine master and containing, apart from a perfect technical treatment, invaluable historical vignettes.

Finally, for exercises proper, an excellent source is Schaum's Outlines series .
The books there are very user-friendly and the exercises quite reasonable, with a progression from very easy to more demanding, accompanied by clear, detailed solutions.
Look here for the dirt cheap volume ($13.41 !) on Complex Variables.

share|cite|improve this answer
Thank you for sharing your experience. This was very helpful. I'm actually enjoying the book so far. I found Riesz representation theorem and how Lebesgue measure was constructed just beautiful. However, chapter 2 exercises really frustrated me. I'll check your book recommendations for better exercises. – PeterM Sep 25 '12 at 18:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.