A few words on the book “Functional Analysis” by Peter Lax

by Orr Shalit

I recently bought Peter Lax‘s textbook on Functional Analysis, with a clear intention of having it become my textbook of choice. I heard nice opinions of it. Especially, I thought that I would find useful Lax’s point of view that gives the Lebesgue spaces L^p the primary role, and pushes the Lebesgue measure to play a secondary role (I wrote about this subject before).

In fact, the first time I heard of this book was in the following MathOverflow question, that is likely to have been triggered by Lax’s comment (p. 282) “[It is] an open question if there are irreducible operators in Hilbert space, and it is an open question whether this question is interesting” (to spell it out, Lax is making the remark that it is an open question whether the invariant subspace problem is interesting!). I have nothing against the invariant subspace problem (of course it is interesting!), but I was sure that I would love reading a book by a mathematician with a bit of self humor (it turns our Lax’s remark appears at the end of a chapter devoted to invariant subspaces).

In one sense the book was a disappointment, in that I realized that I could not, or would not like to, use it as a textbook for courses I teach. I really don’t like its organization, and I don’t love his style. And the pushing back of Lebesgue measure is a very minor topic (which makes good sense because this is a textbook that was used for second year graduate students). I will have to write my own lecture notes.

On the other hand, the book contains a lot of very neat applications of functional analysis (I won’t spoil it for you, but some are really fun!), and so much better to have it coming from someone like Lax. That’s enough to justify the purchase.

But mathematics aside, this book will now stay close to my heart and change the way I approach the subject of functional analysis. This is because of several historical notes dotted throughout the book. Here is an example, which caught me completely unprepared at the end of Chapter 16 (p. 172) (I read the book non-linearly):

During the Second World War, Banach was one of a group of people whose bodies were used by the Nazi occupiers of Poland to breed lice, in an attempt to extract an anti-typhoid serum. He died shortly after the conclusion of the war.”

And so, when reading through the book, we meet some of our familiar (and also not-so-familiar) heroes of functional analysis being deported to concentration camps, or committing suicide knowing what awaits them from the hand of the Nazis, or somehow making it safely to the west. Some other players are involved in the race to construct a nuclear bomb, or to crack the Enigma code (as I learned from the book, Beurling also broke this code – well, we all knew that he was very smart, but this was surprising. By the time we reach the chapter on Beurling’s theorem, we are not surprised that Lax cannot just leave the anecdote there without mentioning also Turing’s tragedy).

I realize that I rarely connected the history of mathematics and the history of Europe in the twentieth century. It is an unusual and disturbing but also a good thing that Lax – who was born in the nineteen twenties in Hungary and lived through the war in the US – makes this connection. I find it very strange that I always knew well, for instance, that Galois died in a duel, but I never heard that Juliusz Schauder was murdered by the Nazis; I use the open mapping theorem all the time.

Thank you, Peter Lax.

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