Topological vector spaces book recommendation

Question

I'm currently taking a class covering the theory of topological vector spaces using the book Topological Vector Spaces, Distributions, and Kernels by Francois Treves. I find the subject to be very interesting, but its also been quite difficult for me to understand some of the material or do some of the exercises. The course aims to cover most of the first part of Treves book, basically up to Frechet spaces or LF spaces.

Are there any other books that cover roughly the same material as in Treves book that might be a bit easier to go over? I've already checked out other books by H.H. Schaefer and M. P. Wolff, G. Kothe, and Bourbaki, but I've found all these books to be more difficult than the Treves book.

My main interests in topological vector spaces are on the theory of distributions, functional analysis, and applications to partial differential equations.

Thanks in advance.

Answer 1

I think you may find useful chapters 4 and 5 of Alpay, An Advanced Complex Analysis Problem Book. Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions. It is at the same level as Treves' classic book. A strong point of Alpay's text is that - since you are struggling a bit with the main concepts of the theory - it contains exercises with worked solutions.

Hope it helps.

Answer 2

I am not sure what your level is (whether you are comfortable with Rudin's functional analysis and Evans). I think Conway's book might be a good reference on topological vector spaces, though a lot of important concepts are not covered. If you find Conway's book to be too easy or too boring for you, then maybe Rudin's book is a better choice. There are plenty of PDE experts in the forum, so I am sure you can get more input from them. If you seek a golden reference from a fields medalist, then you can try Grothendieck 's book, which might be outdated and dense, but addressed topics often not found in modern TVS textbooks.

Update:

I was clearly ignorant, but I found Roberston&Roberston to be really delightful to read. In particular it has a nice chapter on projective tensor products for Frechet spaces, which is often absent from other sources.