Noncommutative Analysis

Category: Books

Tapioca on page 49

To my long camping vacation this year I took the book “Topological Vector Spaces” by Alex and Wendy Robertson. I “inherited” this book (together with a bunch of other classics) from an old friend after he officially decided to leave academic mathematics and go into high-tech. The book is a small and thin hard-cover, with pages of high quality that are starting to become a delicious cream color.

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A list of corrections to my book

My friend Daniel Reem has encouraged me to put up a webpage with corrections to my book “A First Course in Functional Analysis” (in fact, he even convinced me to insert a promise to put up such a page in the preface to my book, so that I don’t really have an option not to put up that corrigendum). Daniel was also kind enough to find a couple of mistakes! So thanks to Daniel, the page is up: here is the link.

Anyone who wishes to report mistakes can use the comment section in this page, or email me.

A review of my book A First Course in Functional Analysis

A review for my book A First Course in Functional Analysis appeared in Zentralblatt Math – here is a link to the review. I am quite thankful that someone has read my book and bothered to write a review, and that zBMath publishes reviews. That’s all great. Now I have a few words to say about it. This is an opportunity for me to bring up the subject of my book and highlight some things worth highlighting.

I am not too happy about this review. It is not that it is a negative review – actually it has a rather kind air to it. However, I am somewhat disappointed in the information that the review contains, and I am not sure that it does the reader some service which the potential readers could not achieve by simply reading the table of contents and the preface to the book (it is easy to look inside the book in the Amazon page; of course, it is also easy to find a copy of the book online).

The reviewer correctly notices that one key feature of the book is the treatment of L^2[a,b] as a completion of C([a,b]), and that this is used for applications in analysis. However, I would love it if a reviewer would point out to the fact that, although the idea of thinking about L^2[a,b] as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with L^2 in this way (i.e., without requiring measure theory) all the way up to rigorous and significant applications in analysis. Moreover, it would be nice if my attempt was compared to other such attempts (if they exist), and I would like to hear opinions about whether my take is successful.

I am grateful that the reviewer reports on the extensive exercises (this is indeed, in my opinion, one of the pluses of new books in general and my book in particular), but there are a couple of other innovations that are certainly worth remarking on, and I hope that the next reviewer does not miss them. For example, is it a good idea to include a chapter on Hilbert function spaces in an introductory text to FA? (a colleague of mine told me that he would keep that out). Another example: I think that my chapter on applications of compact operators is quite special. This chapter has two halves: one on integral equations and one on functional equations. Now, the subject of integral equations is well trodden and takes a central place in some introductions to FA, and one might wonder whether anything new can be done here in terms of the organization and presentation of the material. So, I think it is worth remarking about whether or not my exposition has anything to add. The half on applications of compact operators to integral equations contains some beautiful and highly non-trivial material that has never appeared in a book before, not to mention that functional equations of any kind are rarely considered in introductions to FA; this may also be worth a comment.

The preface to “A First Course in Functional Analysis”

I am not yet done being excited about my new book, A First Course in Functional Analysis. I will use my blog to advertise my book, one last time. This post is for all the people who might wonder: “why did you think that anybody needs a new book on functional analysis?” Good question! The answer is contained in the preface to the book, which is pasted below the fold.

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