# Functional analysis text for junior-senior math majors?

I am looking for a good book for junior-senior US math majors on functional analysis. Some background and detail:

Last semester I taught a special topics course in functional analysis to junior-senior level math majors. It was the first time teaching functional analysis both for me and for my institute as a whole. I used the book Linear Functional Analysis by Rynne and Youngson, partly because I like the overall structure and partly because my students can get Springer e-books for free through my institute's library, and as much as possible I try to support and facilitate free, open information.

Rynne and Youngson is a nice book, but the issue I encountered is that for most US undergraduates, it assumes more preparation than they are likely to have (I think it is really meant for UK undergraduates, who specialize sooner). For example, it assumes you already know what $L^p$ spaces are and just need a quick reminder. My students had had one semester of real analysis at most, so I ended up writing a lot of material for them on metric spaces, measure theory, Lebesgue integration, etc.

For another run of the course, I am looking for an alternate book I could use instead, that would include a more detailed treatment of background information. It would also be good for the book to be somewhat geared toward the application of functional analysis to PDE, though the students last semester loved learning about the more pure aspects of infinite-dimensional Banach spaces, so it doesn't have to be super applied.

Recommendations? I saw mention on this site of Kreysig, which looks good but I am slightly worried about availability after looking on Amazon. A nice bonus would be if the book has an e-version I can get the library to buy so that students can use it for free.

• I seem to recall Kresig has a cheaper international version... yes see: abebooks.com/servlet/… for example. – James S. Cook Aug 30 '16 at 23:01
• I haven't read Rudin's Functional Analysis, but if this were my project it's the first thing I'd check out. – Daniel R. Collins Aug 31 '16 at 3:43
• The names "Rudin" (maybe first "real and complex analysis" for integration and $L^p$ space, then probably "Functional analysis") and "Brezis" come to my mind, but I haven't time to (re)check them. – Benoît Kloeckner Aug 31 '16 at 8:57
• @JamesS.Cook thanks! Good point! That's definitely a lot more affordable... – Idempotent Aug 31 '16 at 13:26
• @BenoîtKloeckner and DanielR.Collins Thanks! I too am a big Rudin fan. My thinking was that Rudin FA might be a bit too formal for my students at this stage - reading it might feel to them the way reading Bourbaki feels to me! - but I like the idea of making a hybrid between Rudin RCA and Brezis, a book I only recently discovered but am very enamored of. – Idempotent Aug 31 '16 at 13:30

I am, perhaps, late to the party, but this question has no answer, so here are a couple of my own suggestions:

1. Stein, Elias M.; Shakarchi, Rami, Functional analysis. Introduction to further topics in analysis, Princeton Lectures in Analysis 4. Princeton, NJ: Princeton University Press (ISBN 978-0-691-11387-6/hbk; 978-1-400-84055-7/ebook). xv, 423 p. (2011). ZBL1235.46001.

There is a five volume set by Stein and Sharkarchi that covers nearly all of the material that a group of clever undergraduates might hope to see before graduate school. I am most familiar with the books on real and Fourier analysis, but those are both quite good. The one complaint that I have is that the books are a little Janus-headed: there is some excellent exposition, and there are some great problems, but the text and exercises often don't seem to gel entirely. It is sometimes as though one author wrote the text and the other the exercises, without much communication between them. Still, I would recommend the series highly.

2. Kreyszig, Erwin, Introductory functional analysis with applications, New York etc.: John Wiley & Sons. XIV, 688 p. (1978). ZBL0368.46014.

This is the book that I worked out of as an undergraduate. It is a little old, but still quite nice. The exposition is careful and thorough, but not to to point of being pedantic. It is also available fairly cheaply, though you have to look for it.

3. Reed, Michael; Simon, Barry, Methods of modern mathematical physics. 1: Functional analysis, New York-London: Academic Press, Inc. XVII, 325 p. (1972). ZBL0242.46001.

I would hold this text up as the gold standard in functional analysis. The exposition is clear and concise without being overly terse, and the exercises are generally quite good. It may be a bit advanced for an undergraduate audience, but I would still recommend checking it out. I could imagine teaching a very satisfactory quarter (or semester) long course out of the first three or four chapters. That being said, you should gauge your audience carefully. I don't think that it requires that much background, but it starts of fairly quickly with some measure theory and moves on from there. Your students would need to be fairly motivated and committed.

4. Rudin, Walter, Functional analysis, McGraw-Hill Series in Higher Mathematics. New York etc.: McGraw-Hill Book Comp. XIII, 397 p. (1973). ZBL0253.46001.

There is always Rudin. Personally, I wouldn't use this text. I find Rudin's texts incredibly helpful and useful to have around as references—they are quite encyclopedic in that regard—but I wouldn't want to teach a class from any of his books. Rudin's style is terse to the point of perversity, and he seems fond of pulling rabbits out of his hat (I can't think of anything in the functional text off the top of my head, but the proof of the mean value theorem in Baby Rudin is, to my mind, particularly egregious). Moreover, Rudin works in a much more general setting than is likely to be reasonable for undergraduates (as with Reed and Simon). On the other hand, if you have a particularly ambitious group of students or if you are willing to put in a lot time to prepare them, Rudin can be a good reference or secondary text to supplement lectures notes.

• Thank you for such a thorough answer! I too am a big fan of Reed and Simon - I have never encountered it in a classroom setting but I use it constantly as a reference. After hearing about Kreyszig from the comments on my question, I got it and agree it is a really nice book. – Idempotent Dec 12 '17 at 21:58
• @Idempotent Fun story: a colleague of mine needed copy of Reed and Simon for a graduate level functional class. He scoured eBay and found a "well used" copy of the text for cheap. It arrives. It is full of marginal notes, errata on the exercises, and corrections to the bibliography. I mean, notes everywhere. Every bit of the book had been marked and commented on. It turns out that this particular copy had been owned by Arthur Wightman! (Wightman being Barry Simon's Ph.D. advisor.) – Xander Henderson Dec 13 '17 at 3:22

Worth a mention is Pons' Real Analysis for the Undergraduate: With an Invitation to Functional Analysis. While it is primarily an introductory real analysis text, the final section of each chapter is a functional analysis topic. For students who have already had a light introduction to real analysis, you could still make use of these final sections and possibly chapters 8 and 9 on Lebesgue measure and Lebesgue integration respectively as a bridge to further material. Furthermore, the book is in the Springer e-books collection and therefore has the same advantage as Rynne & Youngson in being available for free to your students.

• Thank you! I hadn't heard about this text. The Springer e-book availability is a huge boon! – Idempotent Dec 12 '17 at 21:59