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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.

Asaf Karagila
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PeterM
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    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
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    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

1 Answers1

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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.

Georges Elencwajg
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  • 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