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)

I am an algebraist and not an analyst, however my favourite book on this area is "Walter Rudin: Functional Analysis". 


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". 


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. 


Some of the good books are:
$\textbf{Note.}$ 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. 


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


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. 


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). 


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. 


Zimmer's Essential Results of Functional Analysis is a very interesting read, specially if you already know some basic stuff in functional analysis. 


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. 


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. 


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). 


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'. 


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 


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. 


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 


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. 

