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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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You should try out Rudin - Real and Complex Analysis, in my opinion this is a very nicely written book and I enjoyed it. – anon Oct 22 '10 at 16:35

15 Answers 15

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).

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Fair enough. I have not looked at Baby Rudin yet (My undergrad analysis course was a Moore method course). On the linear algebra front, I have done Strang's Introduction to Linear Algebra (most of it). I plan to move on to something more abstract probably "Linear Algebra Done right". This may or may not be useful but here is what the two course sequence that I will be taking looks like:… . I have taken a look at the course notes followed and the background outlined in the first chapter were rather unfamiliar. – EVK Oct 22 '10 at 18:05
@EVK I think any instructor that tries to teach an undergraduate real analysis course via the Moore method is a sadist.Even if the students are good enough to handle it,I just can't see a lot of material getting covered. – Mathemagician1234 Mar 29 '12 at 6:53
Hi @WillieWong, what about measure-theoretic real analysis? Or, do you think that stuff, e.g., Lebesgue measure and integration, can just be picked up along the way, during the Functional Analysis course? Thanks, – User001 May 12 at 20:41
@User001: for the beginning parts of functional analysis it is not 100% necessary to know measure theory. On the other hand, spaces of integrable functions provide a host of examples of function spaces with different properties, so it can be good to know that. – Willie Wong May 12 at 20:55

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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+1: I'm using Kreyszig and the book is just so good!! – Rudy the Reindeer Oct 30 '11 at 8:42
One of the best books at introductory level =D – PtF Feb 18 '14 at 18:26

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.

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Ah. Okay. I am kind of surprised because I was told that measure theory would make more sense if I learned some functional analysis first (which was one among many of the reasons why I wanted to read up more functional analysis). I have also been recommended Rudin's "Functional Analysis" and Komogorov's "Elements of the theory of functions and functional analysis". However, the recommender didn't appear to have more information as to whether either would be too much to chew on for a first bite. – EVK Oct 22 '10 at 17:22
Rudin did also write a book on Functional Analysis, I wonder if that was what muad meant. – Willie Wong Oct 22 '10 at 17:23
@Eshwaran: if you plan to go the Rudin route, you almost necessarily need to go through his Real/Complex analysis book first. The particular presentation that Rudin gives in his functional analysis book requires a certain familiarity with real and complex analysis, which is covered at least in his book on that subject. – Willie Wong Oct 22 '10 at 17:25
@Willie Wong, no - I meant it - best start with a beginner book and you can just ignore the complex analysis stuff at the end. The book I recommend introduces measure theory in chapter 1. And the most important thing: If you don't like it you can put it down! – anon Oct 22 '10 at 17:41
Whether you need measure theory or topology is debatable, depending on how the course is going to work. I took my first functional analysis class with neither of these as prerequisites, just linear algebra and analysis, and it was fine; the lecturer took a self-contained approach to the L^p spaces due to Mikusinski. Even if you don't do that it's possible to do functional analysis on l_p(Z) and spaces of continuous functions without knowing any measure theory at all (again, as long as your linear algebra and real analysis are solid). – Qiaochu Yuan Oct 23 '10 at 9:33

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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From what (little) I've read from Kreyszig, he's rigorous! Luckily, applied doesn't always mean non-rigorous... Also luckily, rigorous doesn't always mean dense reading :P – Bruno Stonek May 11 '11 at 3:09

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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I will look at that too. Thanks. – EVK Oct 23 '10 at 4:40

I am also an EE. And I think Kreyszig's Funcational Analysis is a good book for your background.

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OK, Thanks. I should probably take a look at that. I know that he had written a book on the subject but I didn't take a closer look because his engineering mathematics book left a bad taste in my mouth during my undergrad. – EVK Oct 23 '10 at 4:37

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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I think the book makes the case very well that economists are not really practical people. I guess engineers would prefer a book in which not all vector spaces are assumed to be real. – Michael Greinecker Apr 8 '12 at 11:42

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.

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+1 for probably the single best introduction to the subject for self-study and strong undergraduates.Anyone struggling with more advanced texts should try this one first. – Mathemagician1234 Mar 25 '13 at 17:41

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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I would suggest 'Functions, Spaces, and Expansions: Mathematical Tools in Physics and Engineering' by Ole Christensen. A very good book. And the bonus is... there is a lecture series based on the book by him that you can watch at youtube. Wow... his lecture is superb. Check this out

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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 like Zeidler's Applied Functional Analysis (Volume 1 and Volume 2).

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In my humble opinion "Funktionalanalysis" by "Dirk Werner" is one of the best books.

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Is is in German? :) – H. R. Jul 18 at 21:47
Yep and it is very good!! – PtF Jul 19 at 20:12

You can read the book "A Course in Functional Analysis" by John B. Conway.

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I have edited your answer. I am sorry to tell you that, you need to see if your recommendation has been offered by someone previously. If so, you can remain at peace upvoting that answer. But, if you have a nicer reason to read that book, you must write your recommendation as a new Answer. Also consider writing your answers in proper grammatical Structure and correct spellings! – user21436 Feb 12 '12 at 18:16

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