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I want to know if there's any book that categorizes problems by subjects of Functional Analysis.

I'm studying Functional Analysis now a days and I really need to solve some problems in order to assure myself that I've really understood the concepts and definitions.

For example: problems related to the Hahn-Banach theorem or Banach Spaces or Hilbert Spaces or related subjects.

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You might also check out "Banach Algebra Techniques in Operator Theory" by Douglas. The book extends beyond the material of a first course in functional analysis, but the first chapter (on Banach Spaces) and the third chapter (on Hilbert Spaces) cover the basic theory in detail from scratch. Both chapters have a huge and excellent collection of problems at the end. The fourth chapter has, in my opinion, the best introductory treatment of spectral theory around (and the best collection of exercises at the end), and the fifth has an excellent treatment of compact operators and index theory. – Paul Siegel Jun 25 '10 at 20:40
Thanks Paul. I have this book and I have to say that it's a fantastic one. – Axiom Jun 27 '10 at 6:39
I looked for books at Amazon, and found that "functional analysis" is a topic in psychology, with more books than the same-name topic in mathematics... – Gerald Edgar May 25 '15 at 14:00

10 Answers 10

up vote 23 down vote accepted

Another classical book is Theorems and problems in functional analysis by Kirillov and Gvishiani.

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This is exactly what I was searching for. Thanks. – Axiom Feb 11 '10 at 19:20

MR0675952 (84e:47001) Halmos, Paul Richard A Hilbert space problem book. Second edition. Graduate Texts in Mathematics, 19. Encyclopedia of Mathematics and its Applications, 17. Springer-Verlag, New York-Berlin, 1982. xvii+369 pp. ISBN: 0-387-90685-1

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I would have proposed the same book. It's really excellent! – Martin Brandenburg Jan 31 '10 at 14:02
Indeed a classic. However, I haven't looked at it in years. I wonder if it becoming dated? (Of course mathematics books generally age well, but …) – Harald Hanche-Olsen Jan 31 '10 at 15:00
Halmos' book is indeed excellent, but it deals only with operator theory on Hilbert spaces. – Bill Johnson Jan 31 '10 at 19:36

If you can refrain yourself from looking at the hints (which are almost complete solutions for the most part), Functional analysis and infinite-dimensional geometry By Marián J. Fabian, et al. is a very good book with lots of exercises.

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Also Fabian, Habala, Hájek, Montesinos, Zizler: Banach Space Theory, The Basis for Linear and Nonlinear Analysis has a lot of exercises. – Martin Sleziak May 25 '15 at 8:29

I realy like the exercises in Gert Pedersen's book Analysis Now.

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Seconded! Petersen's book is great. – userN Jun 25 '10 at 19:25

P. Wojtaszczyk, "Banach spaces for analysts", Cambridge studies in advanced mathematics contains problems from many areas of analysis. The hints in the back make the problems easier without giving everything away.

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While I've come to appreciate Wojtaszczyk's book more since my PhD days, I personally found it more suitable for dipping into (after one had learned the basics) than for learning from. But this is very much just a matter of my personal taste. – Yemon Choi Jan 31 '10 at 19:45
I agree, Yemon. For the basics, the exercises in virtually any book on Real Analysis (e.g. Folland) are fine. Other sources are old Real Analysis qualifying exams, which many departments have on their web sites. – Bill Johnson Jan 31 '10 at 20:43

Also Brezis' book Functional analysis, Sobolev spaces and partial differential equations contains a lot of nice and interesting functional analytical problems.

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Aufgaben und Lehrsätze aus der Analysis, G. Pólya & G. Szegö

Problems and theorems in analysis, G. Pólya & G. Szegö. Translation by D. Aeppli

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I would like to add to the list my favorite book on Classical Functional Analysis:

Dunford, Nelson; Schwartz, Jacob T. Linear operators. Part I. General theory. With the assistance of William G. Bade and Robert G. Bartle. Reprint of the 1958 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988. xiv+858 pp. ISBN: 0-471-60848-3

This book contains a plenty of exercises, which allow to check understanding and much more.

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Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form I. M. Glazman , Ju. I. Ljubic You will learn (finite)functional analysis by solving problems.(not the easiest way..)

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