I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics). What I am looking for is good books that I could understand to go deeper in this areas, what do you recommend? (I can read in Spanish, English, French and German)

8$\begingroup$ Can you be a little more specific in what you are interested in? If you are interesting in spectral theory, the classic 4 volume treatise by Reed and Simon, modern methods of mathematical physics is definitely recommended. ;) $\endgroup$ – Helge Aug 9 '11 at 1:50

2$\begingroup$ You seem to be getting plenty of answers anyway, but I for one don't know what a "credit" means in your country. Maybe you could list the most advanced topics in functional analysis and topology that you've covered in class; then we'd all know what it would mean to "go deeper". $\endgroup$ – Tom Leinster Aug 9 '11 at 12:15

3$\begingroup$ In class we covered Banach spaces (until HahnBanach theorem), Hilbert spaces (until Riesz Representation theorem) and Lp spaces. Then we did some work (three days) on Banach algebras (how to proof a subspace is dense in them). A credit in the European Union is between 20 and 25 hours of work in class and home. $\endgroup$ – dan232 Aug 9 '11 at 13:05

2$\begingroup$ Not a textbook, but I recommend that you read Dieudonné's History of Functional Analysis if you can find it. Seeing the historical motivation for the basic concepts was quite enlightening for me. $\endgroup$ – Qiaochu Yuan Aug 9 '11 at 14:40

$\begingroup$ Thanks, Dan. In case you don't know, you can edit your original question using the "edit" button below it. And by the way, the credit system is not uniform through the EU: e.g. here in Scotland, a credit corresponds to about 75 minutes of lecture time. $\endgroup$ – Tom Leinster Aug 9 '11 at 15:17
I am an algebraist and not an analyst, however my favourite book on this area is "Walter Rudin: Functional Analysis".

27$\begingroup$ Wow, I guess Rudin is THE right answer! All the more interesting since I think the book is dull as dishwater. $\endgroup$ – Igor Rivin Aug 9 '11 at 2:45

13$\begingroup$ Maybe "an algebraist and not an analyst" wants the shortest presentation possible, so they can return to less painful topics. $\endgroup$ – Gerald Edgar Aug 9 '11 at 14:16

5$\begingroup$ Indeed, while Rudin is useful, he treats many topics (e.g., vectorvalued integrals) as merely obligatory, rather than useful (v.valuedintegrals disappear after their brief sighting). Sobolev spaces are horribly shorted, and treated clumsily when they briefly appear. Quasicompleteness is ignored entirely. These are not odd, isolated topics. $\endgroup$ – paul garrett Aug 9 '11 at 15:53

5$\begingroup$ Waiting to see the new text, FUnctional Analysis by Paul Garrett! $\endgroup$ – Gerald Edgar Aug 9 '11 at 16:15

19$\begingroup$ Perhaps it would be constructive to have an MO CW question asking us to construct a list of "Good mathematics books that are actually bad" together with reasons for the opinions collected. Certain books, when taken biblically by an advisor for example, can serve to dampen mathematical intuition to a criminal degree. I'm NOT saying Rudin's book has this property (I've found this book a very clear and useful reference recently). I'm not going to ask such a question because it really wouldn't be right to do so...but I'd like to have the list of answers, nevertheless! $\endgroup$ – Jon Bannon Aug 9 '11 at 18:02
I am not an analyst of any sort, so you do not need to listen to me, but I really like Lax's "Functional Analysis".


$\begingroup$ A magnificent book that often gives new insights on classical results. $\endgroup$ – Alekk Aug 9 '11 at 9:37

$\begingroup$ AWESOME book by a master and no list of textbooks in this area would be complete without it's mention. And it's clearly designed for students with some familiarity with the concepts,so it may be just what you're looking for, Dan. $\endgroup$ – The Mathemagician Aug 12 '11 at 18:30
Some of the good books are:
Elements of the Theory of Functions and Functional Analysis by Kolmogorov, Fomin.
Functional Analysis, by F.Riesz and Nagy.
Functional Analysis : Spectral Theory by V.Sunder. Freely available here.
Analysis now. By Gert Kjeargård Pedersen. (As suggested by Theo Buehler at $\textbf{Math.SE}$
Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press. ISBN 0691113874.
Functional Analysis, Sobolev Spaces and Partial Differential Equations (Universitext) by Haim Brezis.
Elementary Functional Analysis by Georgi E. Shilov.
Introductory Functional Analysis with Applications by Erwin Kreyszig.
Notes on Functional Analysis by Rajendra Bhatia. (Hindustan Book Agency.)
Functional Analysis by S.Kesavan. ( Hindustan Book Agency.)
Elementary functional analysis By Barbara D. MacCluer
Functional analysis: an introduction By Yuli Eidelman, Vitali D. Milman, Antonis Tsolomitis (AMS)
Principles of functional analysis By Martin Schechter. (AMS)
You may also want to see this thread: Problem books in Functional Analysis
$\textbf{Note.}$ The books which are written in Italics are the ones which I have read partially. The ones which are not in Italics are the ones which I have come to know (by friends, teachers) are good books in Functional Analysis. Also, I really don't know which publisher actually publishes the book in foreign edition written by Kesavan and Bhatia.

$\begingroup$ @Theo: Thanks for correcting the spelling mistakes. $\endgroup$ – crskhr Aug 9 '11 at 11:53

3$\begingroup$ The Stein, Shakarchi series of books have been great thus far. One thing that I enjoy, at least with their Real Analysis book is that it is not as "clean" as some of the other texts that have been around for so long and therefore forces some extra thinking on the part of the reader. Some might not like this, but as a student it helped me. $\endgroup$ – Andrew Aug 9 '11 at 13:49

$\begingroup$ @Andrew: I heard that their Real Analysis text is much better than the complex analysis one. $\endgroup$ – crskhr Aug 9 '11 at 14:01

1$\begingroup$ I am also not an analyst, hence recommend generally user friendly books rather than "experts only" books. I agree especially with RieszNagy, and would mention also the very readable book Spectral Theory by Lee Lorch. This was the text for a course at Harvard I took with Barry Simon, who was I think then an undergraduate. $\endgroup$ – roy smith Mar 5 '14 at 5:28
You might be interested in "Analysis Now" by Pedersen. A very nice book on graduate level analysis in my opinion. It covers some areas of functional analysis as well.
I would suggest the book by Haim Brezis: Analyse fonctionnelle, theorie et applications. It was recently translated into English and you can find the information for the English translation here.

1$\begingroup$ I haven't seen this book before, but it looks good. :) $\endgroup$ – Anand Aug 9 '11 at 7:23

$\begingroup$ It's a good book (I do have it at home), but it stays quite elementary : the same author covers things more deeply in "Functional Analysis, Sobolev Spaces and Partial Differential Equations" (which is mentioned in the answer I upped). Bonus points for being short and elementary, though ;) $\endgroup$ – Julien Puydt Aug 9 '11 at 17:25

$\begingroup$ @ Snark: I think that's the same book. It's the English translation of the French book ;) $\endgroup$ – Mahdi MajidiZolbanin Aug 9 '11 at 18:37

2$\begingroup$ @MahdiMajidiZolbanin The English version contains the translation of the French book but also exercises for almost every chapter with partial correction and problems (with solutions also). I think that the English translation is actually the translation of the French book + the translation of French book of exercises and problem that has never been published. Also the French edition : 240 pages or so. English edition : 600 pages or so... $\endgroup$ – user37238 Feb 27 '14 at 10:41

2$\begingroup$ THIS IS THE BEST OF ALL BOOKS.IT TEACHES ANALYSIS IN ACTION $\endgroup$ – Koushik Apr 9 '14 at 9:36
Since you read German, my favorite is Funktionalanalysis by Dirk Werner. It's not necessarily comprehensive, but it covers a lot, has extensive historical remarks, and is extremely wellwritten  I find it more readable than most math books in English (my first language).

3

1$\begingroup$ I have been learning a lot of stuff from this book the last weeks and it is much better than I first thought it would be. Definitely something to look at! $\endgroup$ – Konrad Voelkel Sep 1 '11 at 20:01
John B. Conway's "A course in functional analysis" is also pretty decent.
There's no reason to listen to me either, but for delving a bit deeper, you might want to check out T. W. Körner's Fourier Analysis. The book consists of very short (often just a couple of pages) chapters which contain gems like computing the age of the Earth.

1
When I was studying, I was influenced much by K. Yosida's "Functional Analysis".

1$\begingroup$ More operator theory oriented. :) I like to use this book as a reference. To study from front cover to back cover might be difficult. $\endgroup$ – Anand Aug 9 '11 at 7:25

$\begingroup$ I used it as a source book for a course I taught. I found it very wellorganised and thorough. $\endgroup$ – Amritanshu Prasad Aug 9 '11 at 9:07

3$\begingroup$ This book is dense. It covers a lot in one book, but it's not something you can read quickly. $\endgroup$ – John D. Cook Aug 9 '11 at 11:04
Apart from the classics already mentioned (Yosida, Brezis, Rudin), a good book of functional analysis that I think is suitable not only as a reference but also for selfstudy, is Fabian, Habala et al. Functional Analysis and InfiniteDimensional Geometry. It has a lot of nice exercises, it's less abstract than the usual book and provides a lot of "concrete" theorems.
And I'm not sure about it, but I heard there is a spanish translation (the original is of course in english).
I'd recommend the Dunford and Schwartz. It's a classic. It's huge  three volumes. But you don't have to read the whole series covertocover. If you read half of the first volume, you'll learn about as much as reading many other books on functional analysis. Volume 1 alone is big, but it's easy to read for a book on its subject.
I personally like a recent book of Helemskii Lectures and Exercises on Functional Analysis. One of the differences with other books on the subject is that it uses the categorical point of view. The author starts with a very brief introduction to the category theory and uses this language throughout the book. It's a sort of modern core of FA book, with a sidelines to some physics applications and of historic nature, a terse advertisement of the quantum functional analysis and so on (but there is no measure theory, Radon Nikodym theorem etc. which are elaborated in many excellent old textbooks.) Also it gives somewhat broader picture of FA sketching some directions and stating from time to time theorems without proofs 'that every student should know'.

$\begingroup$ I like this one very much as well. The Russians not only know mathematics well, they certainly take pride in teaching it. $\endgroup$ – The Mathemagician Aug 12 '11 at 18:27
Zimmer's Essential Results of Functional Analysis is a very interesting read, specially if you already know some basic stuff in functional analysis.

2$\begingroup$ As many here I was schooled on the Rudin's book. However, for a person like me who doesn't work directly in functional analysis but by the nature of his business (dynamical systems) needs often quickly to refresh his memory I warmly recommend Zimmer's book. Probably an appropriate title for the book would be a "Functional Analysis for a working mathematician". It should be on everybody shelf. $\endgroup$ – Predrag Punosevac Sep 1 '11 at 18:03
The book on Functional Analysis by Meise and Vogt
is quite comphrensive and contains beside standard functional analysis more advanced sections on the theory of locally convex spaces. There is also a German version if you want to improve your German by reading both together.
In a course I'm taking now, we're using Gerald Teschl's "Topics in Real and Functional Analysis". It seems like you may already know the first few chapters, however.
It's quite well written, and is free: http://www.mat.univie.ac.at/~gerald/ftp/bookfa/index.html
My preferred text(s) for functional analysis are
(1) B.V. Limaye, Functional Analysis
(2) P.D. Lax, Functional Analysis
(3) R. Larsen, Functional Analysis
Book by Larsen introduces seminormed linear spaces, which in my opinion has enough generali


$\begingroup$ @Koushik why would you recommend not touching Limaye's Functional Analysis text? I am curious because I was thinking of getting it. $\endgroup$ – Vishal Belsare Apr 21 '15 at 11:28

$\begingroup$ IS there any book limaye ever wrote without mistakes? $\endgroup$ – Koushik Apr 21 '15 at 11:55
I have learnt Functional Analysis from Peter Lax himself. His book is his notes. Exactly the same notes as the ones he handed to us. Every chapter consists of one twohours long lecture in a two semester graduate course on Functional Analysis. There are a few mistakes here and there, but this book is really ALIVE! It is as if you are in a class of Peter Lax! (I should note here that, Lax's book is published a long time after I left Courant, and at that time the recommended textbook was Yosida's book together with Dunford & Schwartz.)
Having said all these, I should add that as an undergraduate student, I had taken two semesters of Functional Analysis which covered a part of Rudin's book. I still use this book sometimes, as some topics are presented in a beautiful way, but I believe that it is far from introductory, as it starts with Topological Vector Spaces, and it takes a while before normed spaces are mentioned.
The Book "Funktionalanalysis: Theorie und Anwendung" from Harro Heuser gives you a very good introduction as well.
I second Reed and Simon's methods of mathematical physics. However, if you are interested primarily in the applications of functional analysis to PDE, for the most part a couple of appendices of Evans' book suffice in my opinion.
Let me add to this list of suggestions the nice and recent book by J. Cerda "Linear Functional Analysis", Graduate Studies in Mathematics, Vol. 116, AMSRSME, 2010.
I found Principles of Functional Analysis By Martin Schechter and Functional Analysis By Bachman and Narici to be good introductions to the subject.