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Are there any good "analysis through problems" type books? I've tried reading analysis books but I literally get bored to death, and, until I manage to concoct a way of transforming a normal textbook into a problem book (maybe by trying to prove all the theorems myself, but that probably requires more math maturity on my behalf and I don't have that yet I think), I am really interested in an analysis through problems book. I know there exist good ones for number theory (Burn's pathway into number theory), linear algebra (halmos' problem book), abstract algebra (clarke's abstract algebra), geometry (prasolov), etc. Any for analysis?

Thanks

Edit: New title - I think it expresses "analysis through problems" better.

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  • $\begingroup$ Have you seen Schaum's Outline of Theory and Problems of Real Variables by Murray R. Spiegel? At the end it covers some Lebesgue integration, but the approach is fairly elementary (i.e. it's not graduate level measure theory), but before this it is mostly undergraduate level real analysis (and lower level than baby Rudin). Also, look at this book and the other books amazon.com says are similar. $\endgroup$ Jul 21, 2015 at 19:33
  • $\begingroup$ I personally like a lot Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form It develops a nice intuition behind most of the functional analysis results. $\endgroup$
    – A.Γ.
    Jul 21, 2015 at 19:40
  • $\begingroup$ You might take a look at "Problems and Theorems in Analysis" by Polya and Szego (two volumes). Maybe old-fashioned, but definitely fits "analysis through problems". A classic. $\endgroup$
    – awkward
    Jul 21, 2015 at 22:24

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One book that you can get for free online is Introductory Single Variable Real Analysis: A Learning Approach Through Problem Solving by Marcel Finan.

One book that I'd particularly recommend if you're looking for really unique and interesting analysis problems is Real Mathematical Analysis by Charles Pugh. To give you a taste, here's an exercise from the topology chapter:

Prove that there is no way to place uncountably many copies of the letter T disjointly in the plane.

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P. M. Fitzpatrick - Advanced Calculus. Here is the reviews.

Fitzpatrick's Advanced Calculus is enough to cover Calculus and Real Analysis, and it includes also many exercises as well as it's a rigorous text and very readable for self-learning as well.

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