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.
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.
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.
Friedlander and Joshi's Introduction to the Theory of Distributions is short, elegant and efficient.
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.
Robert Adams' Sobolev Spaces. Maybe not the best first book, but a very good second book.
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 :)
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.
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.
Many books on PDE or functional analysis (e.g. Taylor's) will have a detailed coverage of distributions.
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.
Gel'fand, I. M. and Shilov, G. E.: Generalized Functions
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.
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.
I liked Functional Analysis by Kosaku Yosida. It is book on functional analysis but oriented to get the applications of it to differential equations.
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.
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.
"Mathematics for the Physical Sciences", Laurent Schwartz, Dover 2008 is a simplified English language book that covers some (maybe even much) of Schwartz's theory of distributions. Very readable, helpful and interesting (also $19.95). The title sounds more general than it actually is--really is focused on distributions, and their applications. Schwartz says in the preface: 'This work is concerned with the mathematical methods of physics'.
I would say Fourier analysis, by Javier Duoandikoetxea, AMS.
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.