Noncommutative Analysis

Good news! The Analyst’s Gambit: A second Course in Functional Analysis, is now published!

About the book: it is designed to serve as the accompanying textbook for the Technion’s graduate course in Functional Analysis. It is not necessarily a second course (in some math departments, this would be the material one finds in a first course in Functional Analysis), but the way that we teach FA at the Technion is that we have a first course in Functional Analysis aimed at undergraduates, which covers primarily the Hilbert space theory and harmonic analysis, and there is a second course, which is a graduate course, with all the standard material on Banach spaces, Banach algebras and a bit of C*-algebras as well.

Who needs another book in FA?

Everybody does, and in particular you do! You can either buy your own copy, find a free copy online, or (if I know you) ask me for a PDF – but most importantly you should recommend this book to your librarian, put a link to this post on facebook and twitter, send it in your family WhatsApp group – anything that will help me get the word out! Thanks!

The text on the back cover explains something about the book, but to get a better sense, take a look at the table of contents and preface:

What’s the story with the catchy title?

Given that the title of my first book was A First Course in Functional Analysis, it seemed like I had not much choice but to call the second one “A Second Course in FA”. However, unlike The First Course, which is really tailored for students meeting FA for the first time (typically undergraduates) and where the entire approach revolves around this being a first encounter, the “Second Course” is designed to serve as the basic graduate course in FA. It centered on a presentation of core material, a not on some eclectic selection of special topics. Thus, calling it just “A Second Course” would miss the point and might drive away readers who are looking for a good (I mean great!) book from which to learn this material. So I added the first part of the title, and I hope it will make some potential readers give this book a second look. There is also a deeper reason for this name – it captures the essence of the course and maybe the subject – I suppose one needs to read the book to get it. There is also a bit of story about this, but it is best told in person.

Did the book become obsolete by the time it got published?

When I write a book I not only write, but also think and read about writing and publishing. In the final stages, I started hearing about AI generated books, some guy on twitter boasted that he wrote a textbook in linear algebra using LLMs in two weeks, and I also heard some ideas about a near future where anyone looking for a book on some topic would ask their computer to prepare one, covering exactly what they want to learn and written so that it is a perfect match for the singular reader it is designed for. Also, I recently finished working on an extensive encyclopedic survey – a genre that I felt was losing its relevance and need as I was putting the final touches on it. All this led to some sad thoughts whether writing is becoming a skill of the past, and what if by the time I finish working on the current book there will be no use for manmade books anymore?

The book also has a couple of fresh applications of functional analysis that do not appear in standard textbooks, and I was somewhat worried that somehow these applications would become standard by the time the book was out.

None of these things happened. Functional analysis didn’t become obsolete, my neat applications didn’t become standard material, and books are not yet a thing of the past.

Ironically, the only little thing in the book that I can already tell aged badly was a footnote that said “Google it”. I thought that this was cute and apt where it was placed, but now I think this phrase will become obscure faster than anything else in the book.

Could I have written a better book?

So, LLMs couldn’t have written such an excellent book, and I don’t think they will be able to, at least not in the next few years. The next question I asked myself is: who would have written a better book? And it then painfully dawned on me that I could have written a better book. As you can tell, I am quite proud of this book, but I realize that there are several things that I could have done better. My biggest regret is that first chapter is the least exciting in the book. It is a quick introduction to (or recap of) some basic facts that we need get hold of before we can do something really interesting. This is unlike my first book, in which the introductory chapter is perhaps the best. I am not sure which is better. Wouldn’t it be nice if all the chapters I wrote were fantastic?

Well, maybe I should give myself a break. After all, I am only human.