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I took functional analysis from Conway's book. I thought it was just as abstract and dry as homological algebra, if not more. I knew of no applications.

Then I learned that standing waves on a drum were related to functional analysis, and then that quantum mechanics (as taught in Dirac's classic book) is nothing but functional analysis, with the structure of the atom and the nature of particles all tied up with functional analysis.

I'd love to see a textbook for functional analysis that proceeds through the theory while giving numerous real-life applications. As of now, Dirac's book on quantum mechanics is the best example I can think of, giving a lot of background theory.

Is there an application-heavy functional analysis textbook?

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    $\begingroup$ This is not a textbook on functional analysis, so this is a comment rather than an answer, but Index Theory with Applications to Mathematics and Physics by Bleecker and Booß-Bavnbek has functional analysis at its core (it's about the Atiyah-Singer index theorem!) but is intended to not only cover the mathematics involved but also its connection to modern physics. $\endgroup$
    – user37
    Apr 16, 2014 at 18:56
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    $\begingroup$ You've been dis-served if you've been led to believe that either homological algebra or functional analysis is "dry and abstract". Sure, anything can be done that way... or not. $\endgroup$
    – paul garrett
    Apr 17, 2014 at 13:06
  • $\begingroup$ @paulgarrett Interesting comment. Perhaps that should be another question, about how homological algebra can be shown to be interesting and useful. I'd be interested in seeing the answers. $\endgroup$
    – Brian Rushton
    Apr 17, 2014 at 13:11
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    $\begingroup$ As an example, math.umn.edu/~garrett/m/v/snake_lemma_gamma.pdf shows how to analytically continue the Gamma function via the snake lemma. $\endgroup$
    – paul garrett
    Apr 17, 2014 at 22:57

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There is Methods of Modern Mathematical Physics by Reed and Simon, which is a 4-volume book which teaches functional analysis, with a focus on operators in Hilbert spaces.

Its main aim is to provide a sound mathematical background for the methods used in quantum mechanics, but it serves well as a textbook on functional analyis (for example, it covers some measure theory, the dual of banach spaces with the Hahn-Banach theorem, locally convex spaces).

The theorems are usually rather brief and might require expansion on the lecture. For instance, I remember a lecture (from volume 4) where we did a proof for 2x90 minutes - the book's version of the proof was only half a page, but iirc, this is a rather extreme example.

To sum up, Reed/Simon is application-centered, but not strictly application-heavy, and it's 'only' application is quantum mechanics.

Another book worth looking into might be Mathematical Methods in Quantum Mechanics by Gerald Teschl - it's influenced by Reed/Simon as well as the German books of Weidmann, but a bit more recent. On his webpage G. Teschl provides a link to the pdf version of the book, if you want to have a peek at it. Like Reed/Simon, it is a proper functional analysis textbook, aimed at applications in theoretical physics.

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    $\begingroup$ @DaveLRenfro: It looks like your comment was intended to OPs question, not to my answer. $\endgroup$
    – Roland
    Apr 16, 2014 at 13:53
  • $\begingroup$ Yes, it probably fits better there. When I saw your suggestion of Reed/Simon, which is rather advanced but probably suitable for the OP, I thought it would be helpful to follow-up with something more useful to someone else seeing this who doesn't have the OP's background. After writing it I thought maybe I should just put it as an answer, but I can't figure out how to delete a comment. I've read about deleting comments, but I can find nothing anywhere on my viewscreen that suggests how to do this and none of the comments I've read about deleting comments said how to do so. $\endgroup$
    – Dave L Renfro
    Apr 16, 2014 at 14:00
  • $\begingroup$ @DaveLRenfro: At the end of your comment, there's a pen. If you hover your mouse just after the the pen, an "X" should appear which allows you to delete the comment. $\endgroup$
    – Roland
    Apr 16, 2014 at 14:15
  • $\begingroup$ Thanks! I've deleted my first comment, which now appears in a slighly altered form as an answer. I'll leave my other comments since they could be useful to others. (Rhetorical comment: As a non-mind reader, it would never have occurred to me to hover the mouse arrow over various screen locations looking for hidden messages.) $\endgroup$
    – Dave L Renfro
    Apr 16, 2014 at 14:41
  • $\begingroup$ @DaveLRenfro: I'm not sure if I should delete my first response as well. The way I realized it was possible to delete comments was probably when I wanted to edit my comment and the X appeared. Deleting the somment could be seen as some kind of extended editing, so it makes sense, kind of. But I agree, it would be more transparent with extensive mind-reading skills. $\endgroup$
    – Roland
    Apr 16, 2014 at 21:20
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At the introductory level, Erwin Kreysiz's Introductory Functional Analysis with Applications is excellent. See the reviews at amazon.com. It's probably a bit elementary for you at this point (still, it could be very suitable for others reading this thread for recommendations), but I recommend at least looking through a copy at the library. I've actually had a course using Kreysiz's book, for what it's worth. And a course using Conway's book, by Conway himself, for that matter.

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Peter Lax: Functional Analysis.

It is sometimes difficult to use because of its cryptic style, but it is a great source of applications and a great source of historical references on applications which motivated functional analytic concepts.

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If you know German, then the book by

Harro Heuser

is the one you are looking for. It contains many many applications and mixes the theory with relevant applied material. An English translation of an earlier edition is available from Wiley.

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Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations.

The book is a great introduction into functional analysis but it is built up to show how FA results can be used to solve partial differential equations. So if you count PDEs as applications, then this is a great book.

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