Can anyone recommend a book(s) that covers these topics: Measure theory / lebesgue integration Hilbert Spaces Distributions PDE's The only material I have is the lecture notes and they are quite difficult to work through. I need to get the basics I think, before I will understand anything. I find all of this quite abstract and can't really understand the point of all these things we are constructing. thanks
Hi Markel! I doubt you will find a good book that covers all those topics. But for the measure theory/integration part I highly recommend "Lebesgue Integration on Euclidean Space" by Jones. It is extremely well-written and it has a great selection of exercises. Furthermore, it presents measure theory from a down-to-earth, practical viewpoint.
For Hilbert spaces, I strongly recommend Young's Introduction to Hilbert Spaces . I thought the book was very easy to read, and believe it is wonderful introduction to the subject.
All of these topics were covered in a single (one-semester? grad? undergrad?) course?! As micromass pointed out but Folland"s grad text Real Analysis: Modern Techniques and Their Applications http://www.amazon.com/Real-Analysis...=sr_1_1?s=books&ie=UTF8&qid=1309088637&sr=1-1 might come close. I am not sure that it is at the level you want, though. Another book by Folland, Fourier Analysis and Its Application http://www.amazon.com/Fourier-Analy...=sr_1_4?s=books&ie=UTF8&qid=1309088637&sr=1-4 has a nice introduction to distributions. I am also unsure of the presentation of measure used in your course. micormass gave a concrete introduction that has very good reviews. As an undergrad student, I used Bartle's The Elements of Integration, and I quite like it. It gives a short, readable introduction to abstract measure and integration theory, and It has been bundled together with The Elements of Lebesgue Measure and reissued as The Elements of Integration and Lebesgue Measure, http://www.amazon.com/Elements-Inte...2226/ref=sr_1_3?ie=UTF8&qid=1309086750&sr=8-3.
I second the recommendations for the books by Jones and Bartle. For measure, integration, and Hilbert spaces, I also like Real Analysis by Bruckner, Bruckner, and Thomson. Rudin's Real And Complex Analysis is well worth a look, too. For distributions, there's a relatively new book that looks really good but which I haven't read yet: Distributions: Theory and Applications. Or, you could go with Knapp's two-volume set Basic and Advanced Real Analysis, or Rudin's Functional Analysis. Knapp's pair of books comes as close as I've seen to covering all of the topics you listed, and many other topics as well.
Yes, Undergraduate. 4th semester. lol. I'm studying in germany and the mathematics courses seem to be advanced to the point of absurdity. All topics are treated with the upmost rigour and very few examples are given so I'm very thankful for everyones sugestions. I'm mainly interested in a book that would cover the material at a level below what is being taught in the lecture. Anyway, my exam is in a few weeks so most likely I will be taking these books for self study over the summer. Thanks again for all your input!
i recommend sterling berberian's books Fundamentals of real analysis, and introduction to hilbert space for the first two topics on your list. they are clear and easy to read. I have never read a book on distributions, but they are very useful objects. the basic idea is the rule of integration by parts. i.e. if a smooth function g vanishes at infinity, then the integrals of fg' and -f'g are equal. Therefor even if you cannot define f', you can still define the integral of f'g for all smooth g vanishing at infinity. since a function is determined by its integrals against all such functions, this uniquely determines f', even if f is not a differentiable function in the usual sense. This clever theory allows us to find candidates for solutions of differential equations by looking among functions not known in advance to be differentiable. afterwards we want to find genuinely differentiable functions which equal those putative solutions. Those are called smoothing theorems, or something like that. anyway a "distribution solution" to an equation is something that integrates against smooth functions just as an actual solution would do. then you have to prove that for some specially interesting equations, every distribution solution comes from an actual solution. breaking the solution process down into these two steps makes it easier to find solutions, because the space of distributions is closed under limits better than the space of actual differentiable functions. arnol'd has some nice books on ode and pde.
I don't know if it is what you are looking for, but a significantly lower level introduction to functional analysis, that begins with distribution theory, Fourier transforms and PDEs (but contains no measure theory!) is "Applied functional analysis" by Griffel. It could be read by someone with a good advanced calculus background. To keep things "simple", it reduces the scope, and only considers operators with discrete spectra. The PDEs portion mainly focuses on finding solutions, not on fundamental existence/uniqueness types of questions. I checked it out from the library at work mainly for the distribution theory portion, but have not worked through it in detail. It is super cheap! A higher level book along the same lines is "green's functions and boundary value problems" by Stakgold. It mentions Lebesgue integration and uses a few properties of it, but doesn't dwell on measure theory. It is quite theoretical, but sounds more applied than what you are getting. Your university library probably has a copy. It is very expensive, so I wouldn't buy it unless you have a chance to spend time with it to see if it is what you are looking for. Good luck. jason edit: both of these books have been used in undergraduate applied math courses taught by the math department where I went to school (a reasonably good but not ultra high-level University in US; so NOT MIT, Harvard, Stanford, Princeton, ...) Stakgold is probably most often used in first year graduate applied math courses here in the US; probably never in pure math courses. To me it sounds like you are taking pure math, that could in principle be applied but they aren't interested in helping you make the connections between the theory and the applications ...