# Littlewood on real functions

Speaking of MSRI, I was there for two weeks last month for a summer school on analytic number theory and gaps between primes (co-organized with D. Koukoulopoulos, J. Maynard and K. Soundararajan; the videos are already available online, as well as the exercise sheets prepared by the two TAs, Z. Brady and B. Löffel.) And as usual when visiting an English-speaking town, I spent a fair amount of time in second-hand bookstores (I have developed from my graduate student days the theory that certain categories of books should never be bought new; for instance, can one improve on this old-fashioned cover

of a whodunit by D.L. Sayers, which at least makes it clear to the dimmest reader who indeed did it?)

From Half-Price Books and Black Oak Books, I was well rewarded. In particular, the latter boutique has a rather remarkable selection of mathematical books. I skipped the three copies of Gauss sums, Kloosterman sums and monodromy groups (I got mine for five dollars from the Rutgers University special-deals cart a long time ago), but acquired Freiman’s Foundations of a structural theory of set addition for 7 or 8 dollars (the review by B. Gordon that I link to is quite fun to read), and found a little gem in Littlewood’s rather obscure book The elements of the theory of real functions:

Actually, the obscurity of this book is maybe understandable. It’s rather depressing to read

These lectures are intended to introduce third year and the more advanced second year men to the modern theory of functions

in the preface. But more importantly maybe, despite the promising title, the content of the book has very little to do with functions, real or otherwise. It’s really a semi-rigorous treatment of set theory up to very basic facts about subsets of $\mathbf{R}^n$, that does little to excite or attract attention. (Maybe the citation above reflects the fact that women students in Littlewood’s time were simply too clever to find this of any interest, and prefered to spend their time reading Gödel or Bourbaki?)

Despite the claim further in the preface that Littlewood aimed at excluding as far as possible anything that could be called philosophy, the fluffiness of the statements reminds me strongly of that discipline. Indeed, to see the bland statement

In Prop. 19 take the class to be $O$ itself. We obtain a blank contradiction.

shows that we are not in the most fastidious of company from the purely mathematical point of view.

Part of the charm of this book is the really weird terminology and the dizzying array of apparently pointless notational abbreviations (example at random: Prop. 5, p. 56 states Given any series $S$ of $\Xi_0$ terms, and an $\eta$, $E$, there is a sub-series of $E$ similar to $S$). Of course, the first edition is apparently from 1925, when some poetic license might have been permitted as far as set theory and topology are concerned, but one doesn’t need to be overly formalist to raise one’s eyebrows when understanding (in a “completely revised” third edition of 1954) that a “class” denotes what everyone else calls a set, and that a “set” is what everybody calls a subset of $\mathbf{R}^n$. At least this would go a long way towards explaining the poor track-record of Cambridge Students from that period at having the faintest idea what every mathematician outside England was saying, as far as set theory and topology was concerned. Maybe this was the best way to ensure that they would think about more interesting things?