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Hi all. I'm looking for english books with a good coverage of distribution theory. I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions. Thanks in advance.

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 Community wiki? – Akhil Mathew Apr 4 2010 at 16:51
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 Well, (linear) PDEs and their solutions are the reason why distributions were invented and still are one of the most prominent reason for using distributions today. If you want to do something else with them, perhaps you should say what exactly this is. – Johannes Hahn Aug 27 2010 at 16:01
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 Laurent Schwartz's original treatise Theories des Distributions? Lighthill's Introduction to Fourier Analysis and Generalized Functions? Friedlander's Introduction to the Theory of Distributions? The problem is your qualifier "gentle"... unless you say more about your background and what you hope to glean from the subject, it is hard to give a good recommendation. – Willie Wong Aug 27 2010 at 16:13
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 Also, have you looked at the recommendations in mathoverflow.net/questions/20314/… ? If so, what are they lacking? At least several of the ones recommended in that thread does not have too much of a PDE bias (or that they hide it very well). – Willie Wong Aug 27 2010 at 16:16
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 The other question does look like good place to start. I have one recommendation that I think isn't included in the responses to it, which I posted below. I'm still a bit new here, so I'm not sure how things are run. Shall I delete my answer here and post it over there? Sify, I don't want to do this prematurely since it might inconvenience you. Might there be a difference between the two questions? – Anthony Pulido Aug 27 2010 at 16:53
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18 Answers

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Grubb's recent Distributions And Operators is supposed to be quite good.

There's also the recommended reference work, Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms

The comprehensive treatise on the subject-although quite old now-is Gel'fand, I.M.; Shilov, G.E. (1966–1968), Generalized functions, 1–5,.

A very good,though quite advanced,source that's now available in Dover is Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels That book is one of the classic texts on functional analysis and if you're an analyst or aspire to be,there's no reason not to have it now. But as I said,it's quite challenging.

That should be enough to get you started.And of course,if you read French,you really should go back and read Schwartz's original treatise.

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 +1 for Treves. A very good book. – Johannes Hahn Apr 4 2010 at 17:41
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 There is no substitue for Schwarz "Theorie des Distributions" If you don't read French yet... this book is a good reason to alter that situation. – Charlie Frohman Apr 4 2010 at 18:12
Thanks for the precious suggestions! – Nicolò Apr 4 2010 at 20:35
+1 for Strichartz. It's really beautiful and readable, even for a non-analyist like me. – Andy Putman Apr 4 2010 at 21:16
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 @Maxime When Schwartz wrote his book,the reign of Bourbaki on mathematics in general-and French mathematics in particular-was at it's hieght.Hence,the maximal generality was sought in the exposition. Treves,to a lesser degree,has the same flaw. But these, in my opinion,are still really the best introductions to the subject.I haven't seen Grubb yet,though. – Andrew L Sep 16 2010 at 20:28
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Two very readable, wide ranging and well motivated accounts are "Generalised Functions and Partial Differential Equations" by Georgi E. Shilov, published by Gordon and Breach 1968, and "Advanced Mathematical Analysis" by Richard Beals, published by Springer 1973 (International student edition). Both are unfortunately out of print and I keep hoping Dover will pick them up so I can recommend them. A recent advanced textbook is "Distributions and Operators" by Gerd Grubb, published by Springer 2009 Vol 252 GTM.

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Beals' book I know from finding it by accident in the Queens College library-a rather interesting experimental text that never really caught on. The idea was to design a text that covered graduate level analysis topics using distribution spaces instead of measure and function spaces.This supposedly removed the need for sophisticated tools like measures or functionals and would return both graduate students of mathematics and physics to learn the same coursework.A very original and positive idea-although I'm not sure how successful it would be today. – Andrew L Nov 29 2010 at 9:12
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There's the book by Ian Richard and Heekyung Youn. It describes itself as a "non-technical introduction", which apparently means you don't need to know measure theory, topology, or functional analysis. Nonetheless you do need to think more like a mathematician than a physicist or the like in order to appreciate their approach.

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I'd like to point out a recent (Birkhäuser Cornerstones) textbook on Distribution Theory by Duistermaat and Kolk.

The present text has evolved from a set of notes for courses taught at Utrecht University over the last twenty years, mainly to bachelor-degree students in their third year of theoretical physics and/or mathematics.

(I have followed this course, which was quite fun.)

For a more advanced exposition, Knapp's Advanced Real Analysis is great.

Very complete and advanced (and dry) is Hörmander's The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, which has already been mentioned.

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We just reviewed Duistermaat and Kolk at the MAA Reviews Online and it came very highly recommended by Micheal Berg. – Andrew L Nov 28 2010 at 22:20
The Dutch version had been published many years ago. The book is supposed to be very good. – timur Dec 3 2011 at 19:44
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Why don't people mention about Rudin's book, Functional Analysis. Chapter 1-8 are pretty good for the theory of distribution. The problem is that this book is quite dry, no much motivations behind. So you might have a difficult time in the beginning. It is good to read the book Strichartz, R. (1994), A Guide to Distribution Theory and Fourier Transforms, besides.

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What do you need distributions for? Your request is strange, PDEs are the fundamental application, the origin, and the main source of examples for distribution theory, so no surprise all the books on distributions after a while steer to PDEs.

Thus maybe my advice is misguided since I do not understand your needs. Anyway, in my opinion the best introduction to distributions is a nice little collection of exercises written by Claude Zuily some years ago (Problems in distributions, North Holland). If you finish it you will be familiar with all the basic theory and you'll be ready to delve into the intricacies, which can be challenging (see the first volume of Hormander, which is essentially a treatise on distributions, or the fear-inducing first volume of John Horvath with its fourteen different topologies on spaces in duality :)

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1 
 Zuily's book looks like it'd be fun to teach from. Too bad it is out of print. Thanks for pointing it out. Also, forgive my ignorance, but what is the book of John Horvath's you spoke of? I am not having much luck searching without knowing the title or at least vaguely the subject. – Willie Wong Aug 28 2010 at 0:56
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 The title of the book was Topological vector spaces and distributions. It was the first of a planned 2-volume set, but I guess Horvath got so bored while writing the first one that he never managed to finish the project. I certainly did not manage to finish the book, but for a brief season had in mind the complete topological picture of spaces of distributions. I was a student and was convinced I had to learn from the ground up :) – Piero D'Ancona Aug 28 2010 at 21:06
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For a really gentle introduction I would recommend Kolmogorov and Fomin's Introductory Real Analysis, available as a Dover paperback. They have a nice introduction to distributions as "generalized functions" in Section 21.

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I really like this book too. – Deane Yang Aug 27 2010 at 18:14
Hmmm... I'm not quite sure in which way this introduction differs from all other "distributions are generalized functions"-introductions to this topic. It is the standard approach and as far as I can see every book that was mentioned in the answers and comments to this question uses that introduction. I really would like to know if there is another (useful) approach to distributions. – Johannes Hahn Aug 27 2010 at 20:04
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I agree with Johannes's comment, but despite this, one book that might fit your criteria is Theory of distributions by M.A. Al-Gwaiz. I haven't looked at it for some months, but it made the following standard texts more accessible:

  • Friedlander and M. Joshi's Introduction to the Theory of Distributions.
  • Hörmander's The Analysis of Linear Partial Differential Operators.

A book that I haven't looked at thoroughly, but you might find interesting, is Guide to Distribution theory and Fourier transforms by Robert S. Strichartz. I once took a class with the author, whose verbal explanatory style is complete and who is also a clear writer.

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I liked Functional Analysis by Kosaku Yosida. It is book on functional analysis but oriented to get the applications of it to differential equations.

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Robert Adams' Sobolev Spaces. Maybe not the best first book, but a very good second book.

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and an excellent book to have on one's bookshelf! it seems to get better with age. – Nilima Nigam Jul 12 2011 at 18:08
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If you want a comparatively elementary approach to distribustion theory with applications to integral equations and difference equation no books come close to Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications by A H Zemanian. another plus is it is Dover paperback, so cheap. Check this out. http://www.amazon.com/Distribution-Theory-Transform-Analysis-Introduction/dp/0486654796/ref=cm_cr_pr_product_top.

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Very good suggestion,Man-I completely forgot about that one. – Andrew L Aug 23 2010 at 20:50
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Lieb and Loss, "Analysis" quickly starts with measure theory and after a short break with Fourier transforms, gets on to Distributions. I would imagine this is the fastest way to learn distributions.

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If you want to learn modern analysis and it's applications,whether you're in mathematics or the physical sciences,then Lieb/Loss is a must read PERIOD,Anweshi. – Andrew L Aug 23 2010 at 20:51
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Gel'fand, I. M. and Shilov, G. E.: Generalized Functions

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I benefited a lot from Vol. 4. – Anand Sep 16 2010 at 18:28
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Friedlander and Joshi's Introduction to the Theory of Distributions is short, elegant and efficient.

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I would say Fourier analysis, by Javier Duoandikoetxea, AMS.

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Just my 2c: Being a student with a limited mathematical education, I used V.S. Vladimirov's Generalized Functions in Mathematical Physics (Mir Moscow 1979) and it was not as hard as I expected it to be - Vladimirov was rigorous and pedantic, as a book in mathematics should be, but not too complicated in explaining the concepts.

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One big book on distributions is the first volume of Hormander's The Analysis of Linear Partial Differential Operators. This may not be the easiest book to read, but it is comprehensive and a definitive reference.

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Yikes,that's a scary book. – Andrew L Aug 23 2010 at 20:52
Hormander's books are very challenging... – Anand Sep 16 2010 at 18:25
+1: This is a beautiful and rewarding book. – Andres Caicedo Sep 17 2010 at 2:01
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Many books on PDE or functional analysis (e.g. Taylor's) will have a detailed coverage of distributions.

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