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
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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
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).
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There's a book that could fit your actual level perfectly. The book is Beginning Functional Analysis by Karen Saxe. It is aimed at undergraduates whose background is a basic course in linear algebra and real analysis. It is pretty well suited for self study since it is very readable (I've done it myself), and the author claims that one of its aims is that the book can be used for self study.
It does not require any measure theory because it develops the basics of Lebesgue measure in chapter 3.
I highly recommend it if you want to read a little bit about functional analysis without having to master more things in order to read more advanced books.
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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.
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I am also an EE. And I think Kreyszig's Funcational Analysis is a good book for your background.
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One unconventional book is Infinite Dimensional Analysis: A Hitchiker's Guide by Aliprantis and Border. It's fully rigorous but written for "practical people, such as engineers and economists" rather than math students.
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We used Conway (about 3/4th) of the book for functional analysis. It was very dense and I felt like I wanted to stab myself in the guts, but I made it! ;-).
I agree with the other posts that you need to brush up some material before you do this.
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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.
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Linear Functional Analysis by Rynne and Youngson (Springer Undergraduate) is really understandable if you don't have many prerequisites, comparable to Kreyzig's book. It doesn't cover that much material, but all the basics are there, with all details filled in.
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I would recommend Haim Brezis's Functional Analysis, especially the last version, which contains many exercises and problems. It is more oriented towards the approach of Partial Differential Equations with the aid of functional analysis, but it's definitely one of the best introductions to the subject.
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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.
Not to scare you, but list of requirements for a first course in functional analysis is rather long:
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.