It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see last paragraph)

For someone who has the median knowledge from an "introductory real analysis class," how do you build through these topics?

One option would be to work through 'Real & Complex Analysis' by Rudin before his 'Functional Analysis,' but I am hesitant to commit (dollars) because I have been told he uses 'magic tricks' for the sake brevity while obscuring understanding. Are these people just whiners? The internet is a great supplementary text.

In my sequence we partially worked through Bartle/Sherbert, then the awful pink book by Marsden/Hoffman, if that matters. We went from field axioms & completeness of the reals up through a few simple proofs involving total derivatives and the integral that uses Darboux sums. We avoided discussing actual measure theory by defining a weaker 'volume in $\mathbb{R}^n$'.

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    $\begingroup$ I never liked Rudin's book on functional analysis. Folland's Real Analysis is the standard graduate real analysis book today and covers all the topics you mentioned. Real and Complex Analysis by Rudin is also good. I don't understand your remarks about being "hesitant to commit" -- there's no commitment involved. If you don't like the book, stop reading it. Both books are going to be terse. Graduate level mathematics is hard, so you better get used to it. $\endgroup$ – Potato Jul 22 '14 at 3:36
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    $\begingroup$ @Potato I think you're right about his functional book. He is way too heavy on the topological vector space side of things for my tastes. The material is good but I can really do without the treatise on topological vector spaces. Real and Complex is a much gentler introduction to functional analysis-like ideas by far. Oddly enough though Rudin's Fourier Analysis on Groups is extremely readable. I don't really get it... $\endgroup$ – Cameron Williams Jul 22 '14 at 3:50
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    $\begingroup$ The Bartle/Sherbert textbook is excellent and provides a solid background. The Rudin books, however, are different animals entirely. The book Functional Analysis with Applications by Erwin Kreyszig is fantastic. Look into that book first. (Also, if you are looking for nice book before Rudin's Real and Complex Analysis text, try Bartle's Elements of Integration and Lebesgue Measure.) $\endgroup$ – dgc1240 Jul 22 '14 at 3:52
  • $\begingroup$ You may want to define "introductory real analysis" because that means at least two different things to me. $\endgroup$ – Brad Jul 22 '14 at 4:20
  • $\begingroup$ real analysis in the most introductory-level sense. The italics are what we covered $\endgroup$ – Christian Chapman Jul 22 '14 at 4:22

Kreyszig's book Introductory Functional Analysis with Applications is a classic that might be good at your level.

I like the way the fundamentals of functional analysis treated in this set of lecture notes.

For more a more advanced book, consider Conway's A Course in Functional Analysis.

It's also useful to really have a solid intuitive understanding of linear algebra and topology. Functional analysis is basically infinite dimensional linear algebra, if you interpret "linear algebra" broadly enough.

  • $\begingroup$ I think Conway's book is also very good on an elementary level, he also starts completely from the basics. Furthermore, it covers almost all important topics if you want to go deeper. $\endgroup$ – Daniel Aug 2 '14 at 17:21

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