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I have formally studied functional analysis, both as university courses, and by myself, but this is one area of mathematics I find so huge and complicated, I have a hard time properly getting into it.

With that said, I am mainly interested only in a specific area of FA, and that is rigorous spectral theory on Hilbert spaces, and Banach/$C^*$ algebras. Essentially, math needed to make quantum mechanics rigorous. I know there are some books specifically written about QM from a mathematician's point-of-view but I am purely interested in the background math for its own sake.

I would like to request textbook recommendations on this area with the following points in mind:

  • The textbook should be fairly didactic. Think Lee instead of Kobayashi & Nomizu, to use famous differential geometry texts as an analogy.

  • The textbook should contain a brush-up of the necessary functional analysis background. I specifically would like this to be sketch-y in the sense that one of my main problems about FA is that it is so huuuuuge. If detailed exposition is given, I get lost in the details. All I want is an assessment of basic/important theorems, like, I dunno, closed graph theorem etc.

  • Hopefully the textbook should be light/forgiving on measure theory. I know it is needed for the spectral theorem, but it really should contain a brush-up, or at least useful references.

  • I'd like spectral theory covered on Hilbert spaces and $C^*$ algebras in great detail. It should have stuff on unbounded operators too, since most interesting operators in QM are unbounded operators.

  • It is not necessary at all, but I'm interested in spectral theory on Banach-spaces too. Probably not all the way, since I seem to remember that's very complicated, but just some outlook and basics. This is a very minor point though.

Any recommendations much appreciated.

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    $\begingroup$ Possible duplicate of What is the best reference for Spectral theory? $\endgroup$ – Francois Ziegler Sep 2 '17 at 17:22
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    $\begingroup$ These are very specific requests, so probably you'll have to write this yourself if you want a perfect fit. Maybe Conway's book comes close. $\endgroup$ – Christian Remling Sep 2 '17 at 18:00
  • $\begingroup$ @ChristianRemling Well, recs don't need to strictly adhere to such level of specificism, but I figured this'd help establish what I am looking for. Didactic resource on spectral theory that doesn't assume a wide-level knowledge of functional analysis. I'll be checking out the Conway book for sure. $\endgroup$ – Bence Racskó Sep 2 '17 at 18:14
  • $\begingroup$ Actually, Conway doesn't do much FA review, but he has a book on that too that gets referred to quite frequently. $\endgroup$ – Christian Remling Sep 2 '17 at 18:57
  • $\begingroup$ Echoing Christian Remling, I doubt there is a book that fits all of these criteria. But I highly recommend Gert Pedersen's "Analysis Now." He treats the basic results of FA (but not in a 'sketch-y' manner) and gives an excellent treatment of spectral theory for bounded and unbounded operators and an introduction to the operator algebra theory one uses in quantum mechanics. You would be required to work out many ideas for yourself (via exercises and sometimes terse proofs)--but I found that pretty fun. One could read chapters 3-5 and refer to 1 & 2 as needed to not get bogged down in FA $\endgroup$ – Caleb Eckhardt Sep 2 '17 at 22:31
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Maybe the following book will be useful: http://www.springer.com/us/book/9788847028340 (Spectral Theory and Quantum Mechanics, by Valter Moretti).

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If you're willing to compromise on the operator algebras part then "Introduction to Hilbert Spaces with Applications" is close to optimal. It includes not only a detailed discussion of the spectral theorem in all its glory - with gentle, detailed exposition, all background material including unbounded operators and measure theory, and lots of exercises - but also an excellent chapter on the mathematical foundations of quantum mechanics.

If you're willing to compromise on unbounded operators then you should go with "Banach Algebra Techniques in Operator Theory" by Douglas. It is probably the best and most comprehensive treatment of the operator algebraic approach to spectral theory around, with a thorough treatment of the basics of Banach spaces and Hilbert spaces to boot.

Otherwise I would probably go with Zimmer's "Essential Results of Functional Analysis". It's a short book but the writing is extremely efficient, and so you can plausibly get a handle on the main ideas of functional analysis (including what you want) without making a huge commitment.

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I think I know the perfect reference for your particular needs: "Topics in Functional Analysis" by Barry Simon in "Mathematics of Contemporary Physics" edited by R. F. Streater, Academic Press 1972, pp. 17-76. It basically covers the topics you mentioned without going into too much technical detail.

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My book Mathematical Quantization satisfies most of these conditions.

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