Good book for self study of functional analysis

I am a EE grad. student who has had one undergraduate course in real analysis (which was pretty much the only pure math course that I have ever done). I would like to do a self study of some basic functional analysis so that I can be better prepared to take a graduate course in that material in my university. I plan to do that next fall so I do have some time to work through a book fully. Could some one recommend some good books to start working on this?

Thanks in advance

Solution 1:

Check out "Introductory Functional Analysis with Applications" by Erwin Kreyszig. I have not read it myself, but I have heard great things. Also, in the preface, he writes that Calculus and a familiarity with Linear Algebra are all that's needed as prerequisites.

Solution 2:

Considering the fact that you have only had one undergraduate course in analysis and will be taking an actual functional analysis class, I don't think you actually want to self-study functional analysis. It would be much more useful for you to

• bulk up on your linear algebra
• review your real analysis

Functional analysis is, for a large part, linear algebra on a infinite dimensional vector space over the real or complex numbers. Having a good intuition from linear algebra is essential: you'll know what is reasonable to expect when the dimensional infinities can be controlled (by some sort of compactness), and when they cannot be controlled, what parts of the argument cannot possibly go wrong.

A bit of real analysis is also helpful because a lot of topological notions are introduced in those books, and familiarity with them is necessary. Furthermore, notions involved in the normed/metric spaces, basic notions of convergence and compactness, and many such are used all the time in functional analysis.

Therefore I think you will be better off reviewing the notes for your undergraduate analysis course (or going through Rudin's Principles of Mathematical Analysis) and studying some linear algebra (unfortunately I can't think of any good book to recommend there).

Solution 3:

Not to scare you, but list of requirements for a first course in functional analysis is rather long:

• Basic theorems of metric spaces including, but not limited to:
  • The Baire category theorem
  • $\ell^p$ is complete
  • Arzelà-Ascoli (how else will you show that an operator is compact?)
• Measure theory --- or at least be ready to accept that you have to learn some while reading functional analysis. Because the Riesz representation theorem essentially says that for a big class of "reasonable" spaces, continuous linear functionals and measures are the same. In other words a lot of the theory will make no sense without at least knowing some measure theory.
• Topology. If you want to go beyond Banach spaces and study Fréchet spaces. The continuous dual of a Fréchet space that is not a Banach space is not necessarily metrisable --- and you get to work with multiple different topologies on your spaces (weak, strong, weak-*)

If that doesn't scare you off, I can recommend the information-dense "Introduction to Functional Analysis" by Reinhold Meise and Dietmar Vogt. ISBN 0-19-851485-9.

And when I say dense i mean very dense. It clocks in at a modest 437 pages, yet in a late undergraduate course in functional analysis we covered less than a third of that book (plus some notes on convexity) in a semester.

As for Rudin's Real & Complex Analysis: it's a great book, but I don't know if I'd really call it a book on functional analysis. I'd say it's on analysis in general --- hence the title.

UPDATE: If you find that you need to brush up on real analysis, Terence Tao has notes for 3 courses on his webpage: Real Analysis 245A (in progress at the time of writing), 245B and 245C. Actually I think I can highly recommend the entirety of his webpage.

Solution 4:

If you are EE you should read Kreyszig. You don't need the rigor of Rudin or anything similar, you want to be able to apply it. Every physicist reads Kreyszig. You might also try the Dover reprint of Griffel, Applied Functional Analysis, under $20. It is a nice read for someone with only an undergrad analysis course. My favorite, although you might have trouble with your background, is Applications of Functional Analysis and Operator Theory by Hutson and Pym, if you can find a copy. I learned functional analysis from doing quantum mechanics and then read all of the above books. If you take a grad level pure functional analysis course in your math department without the requisite background you may regret it.

Solution 5:

I am somewhat surprised that no one has suggested Functional Analysis by George Bachman and Lawrence Narci. In the preface, the authors claim that only basic real analysis and linear algebra is presumed; nonetheless, many of the facts required from these subjects are developed in the text. Inner product spaces, including the Riesz representation theorem, normed/metric spaces and topological spaces. Granted, the Hahn-Banach theorem isn't introduced until chapter 11 but the text is mostly self-contained and very easy to read.