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:
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).
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).
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
- Introduction to set theory
- Measure, countable measure
- Outer measure, Lebesgue measure, measurable sets
- Measurable functions
- Stieltjes-Lebesgue measure (very very brief)
- Lebesgue integral
- Egorov theorem, Lebesgue theorem, Beppo Levi's lemma
- Vector spaces, definitions
- Topology of normed spaces
- Banach spaces
- The application of contraction mappings to the solution of the Fredholm and Volterra integral equations
- Pre-Hilbert spaces
- Hilbert spaces
- Compact sets and Arzela-Ascoli theorem
- Linear operators, definitions
- Bounded and continuous operators
- Invertible operators
- Closed operators
- Hahn-Banach theorem
- Compact operators
- The application of the theory of compact operators to the solution of second-order equations (Riesz-Schauder theory)
Spectral analysis of linear operators
- Introduction to the spectral theory of linear operators.
- 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!