Twelve Landmarks of Twentieth-Century Analysis

This is a very erudite presentation of the standout analysis theorems of the Twentieth Century. It is a translation of the French-language book originally published by Calvage et Mounet in 2009. This edition is a revised and expanded translation and adds one new chapter on the Hardy–Ramanujan theorem on the asymptotics of the number of partitions of an integer.

At first glance the book appears to be slanted very heavily towards classical analysis (an impression reinforced by the front cover illustration: a 1920 photo of G. H. Hardy and J. E. Littlewood). Wasn’t the Twentieth Century the beginning of our great age of abstraction? This is where some of the erudition comes in: The book does in fact discuss the general theories, but shows them as outgrowths of specific concrete results.

This is a book of theorems and it talks mostly about specific theorems (it does have a chapter on Banach algebras, which is needed for two theorems, Wiener’s Tauberian theorem and Carleson’s corona theorem). The book presents complete proofs, and in most cases gives both the original proof and the best modern proof. Even better, each chapter has a lengthy collection of challenging exercises that develop the ideas further (there are length hints in the back of the book).

The translation reads very smoothly, with just a few awkward spots such as this one on p. 16: “Well, much as we have not invented much about transcendence since Hermite, he did much the same.” In a few cases the bibliography lists French-language books for which it could have cited the English translation; it even references the French translation of Rudin’s Functional Analysis. Names with umlauts are typeset in the bibliography using the Hungarian double accent (they are correct in the body and the index). Names with accents are listed first in the index; thus Lévy precedes Lagrange and Hölder precedes Haar.

This book was a pleasure to read, and I learned a lot from it (and expect to continue learning as I study it). It is well-written and beautifully typeset. Even in areas that I know well, such as Tauberian theory and analytic number theory, the book presented some new facts and perspectives that I didn’t know.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.