Preface to Analysis I

Universitext Springer-Verlag BerUn Heidelberg GmbH Roger Godement Analysis I Convergence, Elementary functions Translated from the French by Philip Spain Springer Roger Godement Universite Paris VII Departement de Mathematiques 2, place Jussieu 75251 Paris Cedex 05 France e-mail: [email protected] Translator: Philip Spain Mathematics Department University of Glasgow Glasgow G12 8QW Scotland e-mail: [email protected] Cataloging-in-Publication Data applied for A catalog record for this book is avaiIable from the Library of Congress. Bibliographic information publisbed by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detaiIed bibliographie data is avaiIable in the Internet at <http://dnb.ddb.de>. ISBN 978-3-540-05923-3 ISBN 978-3-642-18491-8 (eBook) DOI 10.1007/978-3-642-18491-8 Mathematics Subject Classification (2000): 26-01, 26A03, 26A06, 26A12, 26A15, 26A24, 26A42, 26B05, 28-XX, 30-XX, 30-01, 31-xx, 41-XX, 42-XX, 42-01, 43-XX, 54-XX This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965,in its current version, and permission for use must always be obtained from Sprin ger-Verlag. Violatinns are liable for prosecution under the German Copyright Law. springeronline.com @ Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design & production, Heidelberg 1'ypesetting by the Translator using a 'JEX macro package Printedon acid-free paper 4113142ck-543210 Preface Analysis and its Adhesions Between 1946 and 1990 I had thousands of st udents; in the very economical French system with its auditoria for two hundred people or more, t his was not difficult. On several occasions I felt the desire to write a book which, presupposing only a minimal level of knowledge and a t aste for mathematics, would lead the read er to a point from which he (or she) could launch him self without difficulty into the more abstract or more complicated theories of the xx»cent ury. After various attempts I began to write it for Springer-Verlag in the Spring of 1996. A long-est ablished house, with unrivalled experience in scientific publish• ing in general an d mathemat ics in particular, Springer seemed to be by far the best possible publisher. My dealings with their mathematical department over six years have quite confirmed this. As, furthermore, Catriona Byrne, who has responsibility for author relations in this sector, has been a friend of mine for a long t ime, I had no misgivings at confiding my fran cophone production to a foreign publisher who , though not from our parish , knows its profession superlat ively. My text has been prepared in French on compute r, in DOS, with the aid of Nota Bene, a perfect ly organized, simple and rational American word proc essor ; but it is hardly more ad apted to mathematics than the trad itional typewriters of yesteryear: greek letters, E, f , E have to be written by hand on the printout, something I had been doin g anyway since my first 1946 typewriter. I event ually devised a coding syste m , for instanc e [[alpha]] for greek letters, t hat mad e it easier to tran slat e the NB files into TEX by using glob al commands. But apart from simple formulae in the main text , most of t he others had to be typeset again for t he French version. The excellent English translation has been much easier to do since Dr Spain , who typ es in TEX, had the TEX version of t he French edition. I have t aken this opportunity to make some small changes to the French ver• sion. *** This is not a standard textbook geared to t hose many st udents who have to learn mathematic s for other purposes, alt hough it may help them; it is t he read er interest ed in mathematics for its own sake of whom I have thought while writing. To many of the French st udents and particularly to many of VI Preface the brightest, mathematics is merely a lift to the upper strata of society". My goal is not to help bright young people to arrive among the first few in the ent ry competit ion for the French Ecole polytechnique so as to find themselves thirty yea rs later at the service or at the head of a public or private ent erprise producing possibly war planes, missiles, milit ary electronics, or nuclear weapons'", or who will devise all kinds of finan cial stunts to make t heir company grow beyond what they can cont rol, and who , in both cases, will make at least twenty times as much money as the winner of a Fields Medal does. The sole aim of this book thus is mathematical analysis as it was and as it has become. The fundamental ideas which anyone must know - convergence, cont inuity, elementary functions, int egrals, asymptotics, Fourier series and in• tegrals - are t he subject of the first two volumes. Volume II also deals with that part (Weierstrass) of the classical theory of analyt ic functions which can be explained with the use of Fourier series, while the other par t (Cau chy) will be found at the beginning of Volume III. I have not hesitated to introduce, sometimes very early, subjects considered as relatively advanced when they can be explained without technical complications: series indexed by arbit rary countable sets, the definition and elementary properties of Radon measures in lR or C, integrals of semi-continuous functions and even, in an Appendix to Chap . V, a short account of t he basic theorems of Lebesgue's theory for those who may care to read it at this early stage, analyt ic functions, the con• st ru ct ion of Weierstrass elliptic funct ions as a beautiful and useful example of a sophisticated series, etc . I have tried to give the reader an idea of the axiomat ic const ruct ion of set theory while hop ing that he will t ake Chap. I for what it is: a cont ribut ion to his mathematical cult ure aiming at showing that the whole of mathematics can, in principle, be built from a small number of axioms and definitions. But a full underst anding of t his Ch apter is not an obligatory prerequisit e to an apprenticeship in analysis. The only thing the reader will have to ret ain is the naive version of set t heory - standard operations on sets and functions to which, anyway, he will get used by merely reading the next chapters • as well as the fact t hat, even at t he simplest level, mathematics rests upon proofs of state ments, an old art which, in French high schools and probably elsewhere as well, is in t he process of becoming obsolete because, we are told, learning to use formul ae is much more useful t o most people, or because it is t oo difficult for the many children of t he lower strata of society (I was one in the 1930s) who now flood the high schools . th 1 In XIX century Cambridge, the winners of the Math Tripos would far more often become judges or bishops than scientists . 2 One of the brightest stud ents I have known in thirty-five years is today the head of a holding company that controls, among other things, a chain of supermar• kets. He sells Camembert, shrink-wrapped meat , Tampax, orange juice, noodles, mustard, etc. Ifyou have to choose, this is a more civilised way to squander your grey matter. P reface VII The sequel, in Volumes III and IV, explains subjects which require either a much higher level of abstraction (short introductions t o differential var i• eties and Riemann surfaces, general integration, Hilb ert spaces, general har• monic analysis), or , in t he last Ch ap . XII , a much higher level in computation techniques: Dirichlet series of number t heory, elliptic and modular funct ions, connect ion with Lie groups. While the choice of material in Volumes I to IV represents a coherent and nearl y selfcontained block of mathematics, it const it utes nothing more t ha n one particular view of analysis. Other authors could have chosen other views and, for inst an ce, tried t o lead t heir readers into t he t heory of partial differential equations. I have not even treat ed dif• ferential equations in one variable: one can learn all about them in a myriad of books, and the classical results of t he theory, direct applications of t he general principles of analysis, should pose no serious problem to the st udent who has assimilated these reasonably well. In t he two first volumes - Volumes III and IV are writt en in a much more orthodox fashion - I have firmly emphasised, sometimes with the aid of out of fashion excurses in ordinary language, the ideas at t he basis of analysis, and, in some cases, their historical evolution.

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