This book studies the 20^th century evolution of essential ideas in mathematical analysis, a field that since the times of Newton and Leibnitz has been one of the most important and prestigious in mathematics. Each chapter features a comprehensive first part on developments during the period 1900-1950, and then provides outlooks on representative achievements during the later part of the century. The book will be an interesting and useful reference for graduate students and lecturers in mathematics, professional mathematicians and historians of science, as well as the interested layperson.

This volume grew out of the preparation and achievements of the Symposium organized by the Luxembourg Mathematical Society in June 1992, at Chateau Bourglinster (Luxembourg). We express our best thanks to Monsieur Marc Fischbach, ministre de I 'Education nationale, Luxembourg Monsieur Rene Steichen, ministre delegue aux Affaires culturelles, Luxem- bourg, Academie des sciences, Paris, Academie roya1e des sciences, des lettres et des beaux-arts de Belgique, Section des sciences de l'Institut grand-ducal, Luxembourg, and all lecturers and participants. We hope that this contribution may ease a better understanding of the vitality in mathematics during the first half of the present century and provide clues for the explanation of its influence on later developments. We say our gratitude to persons who assisted us with their advice and help for the technical realization of the volume, especially Mrs. Suzanne D' Addato-Mues, Mr. Fran~ois Gaillard, Mr. Hubert Glesener. We are also grateful to Birkhauser- Verlag for having accepted this publication. Now the book should speak for itself ...