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I would like to ask for some recommendation of introductory texts on functional analysis. I am not a professional mathematician and I am totally new to the subject. However, I found out that some knowledge of functional analysis and operator theory would be quite helpful to my work...

What I am searching for is some accessible and instructive text not necessarily covering the subject in great depth, but explaining the main ideas. I am not searching for a text for engineers, some amount of mathematical rigor would be fine. But I found myself unable of reading some standard textbooks covering in great depth a large amount of issues in theory of Banach spaces, etc. I am looking for something that proceeds to the most important topics (e.g., spectral theory) faster than the most of textbooks, but not at the expense of rigor. I.e., something that covers rigorously the main topics, but concentrates only on the main ideas. Simply an accessible introductory text for a fast orientation in the subject.

Moreover, I would prefer a text that does not require any background in measure theory and similar disciplines.

And another question: is there any functional analysis book that deals primarily with sequence spaces? It need not fulfill the description above.

Thank you for your recommendations!

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I think that it is hard to appreciate functional analysis without some prior background in point-set topology, measure theory, complex analysis, and Fourier analysis. A knowledge of the theory of partial differential equations is also very useful. The reason is that many classical examples of Banach spaces (important objects of study in functional analysis) are function spaces which arise naturally in the areas of mathematics I mentioned (e.g., $L^p$-spaces, $H^p$-spaces etc.). –  Amitesh Datta Jun 28 '12 at 13:46
    
@AmiteshDatta Yes, I definitely agree. But the trouble is that, as I have already mentioned, I am not a professional mathematician, but I have found out, that I need some of functional analysis (especially theory of operators on sequence spaces) for my work. I find it quite hard to learn measure theory etc. from literature by myself (although I would definitely like to). Therefore I am interested mainly in the basic ideas (not technical details concerning integral operators, etc.), or, alternatively, in "discrete functional analysis," i.e., an introduction based on sequence spaces. –  042 Jun 28 '12 at 15:59
    
I disagree slightly with Amitesh. One can learn a surprising amount without touching either measure theory or classical Fourier analysis, although those fields do feed back into functional analysis. –  user16299 Jun 28 '12 at 18:30

3 Answers 3

up vote 5 down vote accepted

I think that Kreyszig'book is a good introduction, though not very short.

Spaces of sequences are classical Banach spaces, and there are books that study their properties systematically. A classic book is this one. But also Larsen's book has plenty of examples from $\ell^p$, $c_0$, $c_{00}$.

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Bollobas's Linear Analysis is short and rigorous but may be slightly too compressed (depending on your background). The second half of Simmons's Topology and Modern Analysis remains, in my view, one of the best "soft" introductions to Banach algebras, the spectrum, and ultimately the Gelfand Naimark theorem for commutative C^*-algebras.

If you don't mind just dealing with Hilbert spaces, then the first two thirds of Young's Introduction to Hilbert Spaces may also be worth a look.

None of these texts require any measure theory.

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+1 for recommending one of the great classic undergraduate textbooks of all time in Simmons.The serious student really should look at ALL George Simmons' textbooks, which are all outstanding. –  Mathemagician1234 Aug 16 '13 at 2:47

The best introductory text I know on the subject is available from Dover books very cheap; it's George Bachman and Lawrence Narici's Functional Analysis. It develops virtually the entire subject of functional analysis from scratch with many examples and good exercises-and it requires only advanced calculus (elementary real analysis) and a good working knowledge of linear algebra to work through.

Also, if you just "need to know some functional analysis" for work, you might want to check out some applied textbooks on the subject, of which there are many.

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