The series "Grundlehren der mathematischen Wissenschaften" (subtitled Comprehensive Studies in Mathematics), Springer’s first series in higher mathematics, was founded by Richard Courant in 1920. The Grundlehren were conceived as a series of modern textbooks. This character is obvious in the first 60-70 volumes, but the series underwent a number of significant changes after World War II. Outwardly, the change was in language. Whereas the overwhelming majority of the first 100 volumes were published in German, the following volumes are nearly all in English. A more important change concerns the contents of the books. The original objective of the Grundlehren had been to lead students to the principal results and to recent research questions in a single relatively elementary and easily readable book. Good examples are van der Waerden’s two-volume Introduction to Algebra or the two famous volumes on Methods of Mathematical Physics by Courant and Hilbert.
Today, it is often no longer possible to start very low and, in one or two comprehensive volumes, end up at the frontiers of current research. This explains why many of the later volumes are both more specialized and more advanced. Nevertheless, most of the books in the series are meant to be textbooks of a kind, with occasional reference works or research monographs. Each book should lead up to current research, without over-emphasizing the author’s own interests. There should be proofs, though not necessarily always complete ones, of all the major statement s enunciated. However, the presentation should remain that of an expository volume. Examples of books that fit this description are the book by Siegel and Moser on Celestial Mechanics, Federer’s Geometric Measure Theory, or Hörmander’s Analysis of Linear Partial Differential Operators. This list could easily be extended but these few examples should suffice to show what the Grundlehren stand for. A very important criterion for whether a book should or not go into the Grundlehren is that of its longevity: a Grundlehren volume should continue to have an impact over many years. The tastes of the editors play a pivotal role in the selection of topics. Topics should be of current mathematical relevance, and not too narrow.
Our characterization of Grundlehren leaves ample room for freedom of style and presentation. Authors are encouraged to follow their individual style, but should always have the interests of the readers in mind when presenting their subject. The inclusion of exercises and historical information is encouraged.
The Grundlehren do not strive for a systematic coverage of all of mathematics. There will be occasional overlaps and, conversely, gaps. However, a systematic effort is made to cover important areas of current interest in a Grundlehren volume once these become ripe for Grundlehren-type presentation.
The Grundlehren series is published in English.
Looking at this description of the character and scope of the Grundlehren series, one will see that we have, as far as we could and as far as the development of mathematics permitted it, remained true to the original idea of the series. Many of the older volumes in the series are surprisingly popular; some of them, although 50 years old or older, have not yet been superseded. Examples are Knopp’s book on Infinite Series or that by Hurwitz and Courant on Complex Analysis.
One should perhaps never advertise a contemporary book as a classic but we are certain that many of the books that have been published over the last few years and that will be published over the next few years will earn this attribute through their use by generations of mathematicians.