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I would like to ask you to recommend me a good modern textbook on functional analysis to refresh what I already know. I am a computer science student and for the last two semesters we've been having a functional analysis course, but, big surprise, some months after I passed the exam I discovered my knowledge of the topic has considerably waned.

My requirements for the book would be:

  1. Easy to read (the lecturer never dared to explain any of the topics to us, and we've been forced to do everything by ourselves. One thing I discovered about the lectures was that the material was presented in a very diverse way, topics sometimes were not connected to the neighbouring topics. Some theorems were brief definitions, some had EXTREMELY long proofs, while the idea of the proof was rather simple - i.e., the theorem about the completion of normed vector spaces. And he never gave us any intuitive examples of what these constructs could be like. So the material was unbalanced. I would especially like that the book has intuitive explanations).

  2. Rather new (it is always bothersome to read books so old they are unaware of the existence of personal home computers. Most of the books we have here were published in the 1970s-80s, if not earlier, and they sometimes mention that some of the described problems could be solved on ЭВМ - an archaic Russian term for a computer. I think that the area of functional analysis is not one to be filled with computational examples, but nevertheless, maybe in the book they could illustrate some points with Mathematica or similar things).

  3. It may be for a complete beginner, or a bit higher level, but not a level such that the ordinary student would feel completely defeated on the second page :)

The topics we learned are (by topics I mostly mean main definitions, some properties, some lemmas/theorems. We never went deeply into them):

Measure and Lebesgue integral

  1. Introduction to set theory
  2. Measure, countable measure
  3. Outer measure, Lebesgue measure, measurable sets
  4. Measurable functions
  5. Stieltjes-Lebesgue measure (very very brief)
  6. Lebesgue integral
  7. Egorov theorem, Lebesgue theorem, Beppo Levi's lemma

Normed spaces

  1. Vector spaces, definitions
  2. Norm
  3. Topology of normed spaces
  4. Banach spaces
  5. The application of contraction mappings to the solution of the Fredholm and Volterra integral equations
  6. Pre-Hilbert spaces
  7. Hilbert spaces
  8. Compact sets and Arzela-Ascoli theorem

Linear operators

  1. Linear operators, definitions
  2. Bounded and continuous operators
  3. Invertible operators
  4. Closed operators
  5. Hahn-Banach theorem
  6. Compact operators
  7. The application of the theory of compact operators to the solution of second-order equations (Riesz-Schauder theory)

Spectral analysis of linear operators

  1. Introduction to the spectral theory of linear operators.
  2. Hilbert-Schmidt theorem (both are brief)

I tried to be specific and detailed in my request.

Thank you in advance!

Added later: I would like to thank you all for useful answers, thank you, guys, you are very helpful!

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3  
I liked Lang's Real and Functional Analysis, but it doesn't have any mention of computers, old or new. :) I am curious why you and your classmates didn't talk to the instructor and ask him to give examples and some sense of the intuition behind what was going on. (Боитесь ли вы докладчика?) –  KCd May 5 '12 at 17:54
    
Peter Lax's recent (2002) book Functional Analysis has very positive reviews on Amazon. –  KCd May 5 '12 at 18:02
1  
These doesn't satisfy your contemporary requirement but I like Kolmogorov & Fomin's "Real Analysis" and Kantorovich & Akilov's "Functional Analysis". –  copper.hat May 5 '12 at 18:21
    
@KCd Lax is very good from what I've read of it,but it's not easy. The OP might find it rough going. –  Mathemagician1234 May 5 '12 at 19:47

3 Answers 3

The book "An introduction to Hilbert space" by N.Young is a good one for beginners and for those who want to refresh functional analysis. It is a very nice read andy as good set of exercises. The book does not go too much into measure theory but covers all the other things you want.

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Barbara MacCluer just finished a beginner's text called Elementary Functional Analysis that looks outstanding. It's reasonably short,has minimal prerequisites and covers all the basics. One warning: It assumes fairly thorough knowledge of measure theory,so you're going to need to learn that first. My favorite source for that material is Angus Taylor's General Theory Of Functions And Integration, available for a song from Dover. These 2 books together should fill your needs very nicely. And cheaply,too!

Here's a review of the MacCluer book: http://page.mi.fu-berlin.de/werner99/preprints/maccluer.pdf

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6  
A -1 rating for absolutely nothing. My anonymous fan club at work again. Get a hobby,please. –  Mathemagician1234 May 20 '12 at 22:43

When I was a student, I liked Principles of functional analysis by Martin Schechter a lot. I also like Rudin's Functional analysis, but it requires a bit more "mathematical maturity".

Added later: I did not notice you mentioned also the Lebesgue integration and measure theory. In my experience, this is usually addressed in a Real analysis (or probability) course, and in this regard my suggestions would be Folland's Real analysis: Modern techniques and their applications, and a bit easier and fun to read, Bressoud's A radical approach to Lebesgue's theory of integration. The latter book focuses more on explaining why one needed Lebesgue's theory in the first place, and hence has a lot of interesting historical details.

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