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: rgodement@aol.com

Translator: Philip Spain Mathematics Department University of Glasgow Glasgow G12 8QW Scotland e-mail: pgs@maths.gla.ac.uk

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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

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Preface

Analysis and its Adhesions

Between 1946 and 1990 I had thousands of st udents; in the very economicalFrench system with its auditoria for two hundred people or more, t his wasnot 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 selfwithout difficulty into the more abstract or more complicate d theories of thexx»cent ury. After various attempts I began to write it for Springer-Verlagin the Spring of 1996.

A long-est ablished hou se, with unrivalled experience in scientifi c publish­ing in general an d mathemat ics in particular, Springer seemed to be by farthe best possible publisher. My dealings with their mathematical departmentover six years have quite confirmed this. As, furthermore, Catriona Byrne,who has responsibility for author relations in this sector, has been a friendof mine for a long t ime, I had no misgivings at confiding my fran cophoneproduction to a foreign publisher who , though not from our parish , knows itsprofession superlat ively.

My text has been prepared in French on compute r, in DOS, with theaid of Nota Bene, a perfect ly organized, simple and rational American wordproc essor ; but it is hardly more ad apted to mathematics than the trad ition altypewriters of yesteryear: greek let ters, E , f , E have to be written by handon the printout, something I had been doin g anyway since my first 1946typewriter. I event ually devised a coding syste m , for instanc e [[alpha]] forgreek letters, t hat mad e it easier to tran slat e the NB files into TEX by usingglob al commands . But apart from simple formulae in the main text , most oft he others had to be typeset again for t he French version .

The excellent English translation has been much easier to do sinceDr Spain , who typ es in TEX, had the TEX version of t he French editi on.I have t aken this opportunity to make some small changes to the French ver­sion.

* * *This is not a standard textbook geared to t hose many st udents who have tolearn mathematic s for other purposes, alt hough it may help them; it is t heread er interest ed in mathematics for its own sake of whom I have thoughtwhile 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 arr ive among the first fewin the ent ry competit ion for the French Ecole polytechnique so as to findthemselves thirty yea rs later at the service or at the head of a public or privateent erprise producing possibly war planes, missiles, milit ary elect ronics, ornuclear weapons'", or who will devise all kinds of finan cial stunts to maket 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 F ieldsMedal does.

The sole aim of this book thus is mathemat ical ana lysis as it was and as ithas become. The fundamental ideas which anyone must know - convergence,cont inuity, elementary functions, int egrals, asymptotics, Fourier series and in­tegrals - are t he subject of the first two volumes. Volume II also deals withthat part (Weierstrass) of the classical theory of analyt ic functions which canbe explained with the use of Fourier series, while the other par t (Cau chy) willbe found at the beginning of Volum e III. I have not hesit ated to introduce,sometimes very early, subjects cons idered as relatively advanced when theycan be explained without technical complications: series indexed by arbit rarycountable sets, the definition and elementary properties of Radon measuresin lR or C, integrals of semi- continuous functions and even, in an Appendixto Chap . V, a short account of t he basic theorems of Lebesgue's theory forthose who may care to read it at this early stage, analyt ic functions, the con­st ru ct ion of Weierstrass ellipt ic funct ions as a beau tiful and useful exampleof a sophisticate d series, etc .

I have tried to give the reader an idea of the axiomat ic const ruc t ion of settheory while hop ing that he will t ake Chap. I for what it is: a cont ribut ion tohis mathematical cult ure aiming at showing that the whole of mathematicscan, in principle, be built from a small number of axioms and definitions. Buta full underst anding of t his Ch apter is not an obligatory prerequisit e to anapprenticeship in analysis. The only thing the reader will have to ret ain isthe naive version of set t heory - standard operations on sets and functionsto which, anyway, he will get used by merely reading the next chap ters ­as well as the fact t hat, even at t he simplest level, mathematics rest s uponproofs of state ments, an old art which, in French high schools and probablyelsewhere 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 ist oo difficult for the many children of t he lower st rata of society (I was one inthe 1930s) who now flood the high schools . . .

1 In XIXth century Cambridge, the winners of the Math Tripos would far moreoften become judges or bishops than scientists .

2 One of the brightest stud ents I have known in thirty-five years is today the headof a holding company that controls, among other things, a chain of supermar­kets. He sells Camembert , shrink-wrapped meat , Tampax, orange juice, noodles,mustard , etc.Ifyou have to choose, this is a more civilised way to squander yourgrey matter.

P reface VII

The sequel, in Volumes III and IV, explains subjects which require eithera much higher level of abst raction (short introductions t o differential var i­eties and Riemann surfaces, general integration, Hilb ert spaces, general har­monic analysis), or , in t he last Ch ap . XII , a much higher level in computationtechniques: Dirichlet series of number t heory, elliptic and modular funct ions,connect ion with Lie groups. While the choice of material in Volumes I toIV represents a coherent and nearl y selfcontained block of mathematics, itconst it utes nothing more t ha n one particular view of analysis. Other authorscould have chose n other views and, for inst an ce, t ried t o lead t heir readersinto t he t heory of partial different ial equations . I have not even treat ed dif­ferential equations in one var iable: one can learn all about them in a myr iadof books, and the classical results of t he theory, direct applications of t hegeneral principles of ana lysis, should pose no ser ious problem to the st udentwho has ass imilate d these reasonably well.

In t he two first volumes - Volumes III and IV are writt en in a muchmore orthodox fashion - I have firml y emphasised , sometimes with theaid of out of fashion excurses in ordinary language, the ideas at t he basis ofanalysis, and, in some cases, their historical evolution. I am not , far from it ,an expert in the history of mathematics; some mathemat icians, sensing t heirend coming, devot e t hemselves to it late in life; ot hers, younger, consider thesubjec t sufficientl y interesting to devote a substant ial part of t heir activityt o it; t hey perform a most useful t ask even from the pedagogical point ofview'' since , at twenty, which I once was, one thinks only of forging aheadwithout looking behind , and almost always without knowing where one isgoing: where and when will one learn? I have myself preferred for a quarterof a cent ury to t ake an interest in a kind of history - science , technology, andarmam ent s in t he xx» century - for which mathematics does not prepareone, though t here are some indirect connections. Nevertheless I have madesome effort to convey to the reader t hat t he ideas and the techniques haveevolved, and t hat it took between one and two cent ur ies for the intuitions oft he Founding Fathers to be t ra nsformed into perfect ly clear concepts foundedon un assailabl e arguments , await ing the great generalisat ions of the xx»cent ury.

Adopt ing this point of view has led me, in these first two volumes, sys­temati cally to eschew a perfectly linear exposit ion, orga nised like a clockworkand only present ing to t he reader t he dominant or a la mode point of view,with asso rted Blitzbeweise, lightning proofs in t he sense in which we speak ofBliizkrieq": one ratifies the result bu t does not comp rehend t he st ra tegy untilsix months afte r the bat tle. At the cost of proving t he same classical resultsseveral t imes over I have tried to present several methods of argu ing to the

3 E.Hairer and G. Wanner , Analysis by It s History (Springe r-New York , 1996) , isa prim e example.

4 Rene Etiemble , a great French specialist of comparative lit erature, once madea study of t he sty les prevailing in various kinds of activities. He came to t heconclus ion that mathematical style was the closest there was to the military.

VIII Preface

read er , and to make clear the necessity of rigour by evidencing the doubtfularguments, and sometimes false results, due to mathematician s like Newton,the Bernoullis, Euler , Fourier , or Cauchy. Adopting this point of view length­ens the t ext palp ably, but one of the ground principles of N. Bourbaki - noeconomies of pap er - is, I think, mandatory when one addresses studentsemba rking on a subject.

The other principle of this same author - to subst it ute ideas for com­pu t at ions - appears even more commend able to me whenever it can beapplied. All the same , one will, inevitably, find calc ulat ions in this book; butI have essent ially confined myself to t hose which, inherited from t he greatmathematici ans of t he past , form an integral par t of the theory and can beconsidered as ideas.

Except occasionally, to round off the text , one will find no exercises here.Working at exercises is indispensabl e when one learns mathematics , and onewill find them in profusion in many other books and specialised collections.The majority of French st udent s, obsessed by the st ring of examinat ionsimposed on them , have a very exaggerated tendency to consider the "lect ures"of little use and that only "pract ical work" and "formulae" count or pay. Theresul t is t ha t t he majority of them are able, up to errors in calculat ion, toint egrat e a rati onal function but incap able of answering questions of a genera lnature, e.g. why is a rational fun ction integrable? To understand a theoremis to be able to reconstruct it s proof. To underst and a block of mathematicsdoes not reduce to knowing how to apply its result s; to underst and a t heoryis to be able to reconstruct its logical st ructure. Every mathematici an knowsthis.

One does not learn analysis or anything else from one single book ; t hereis neither Bible, nor Gospel nor Koran in Mathematics. The fact that thespirit of my book is radi cally different from that of Serge Lang, Undergradu­ate Analysis (Springer, 2nd. ed ., 1997) for example, should not dissuade fromreading it , quite the contrary; even less the books of E. Hairer and G. Wan­ner , Analysis by It s History, Wolfgang Walter , Analys is I (Spr inger, 1992,in German) or Reinhold Remmert , Theory of Complex Function s (Springer­New York , 1991, translation of Funktionenth eorie 1 , 4. Auflage, 1995) , whichI have often used , and cite when I do so. T hese excellent books present nu­merous exercises , as does Jean Dieudonne's Calcul Infinitesimal (Hermann,1968) though his style ent huses me less.

I have not acceded to the new fashion which likes to decorat e elementaryanalysis textbooks with numeri cal calculations to fiftee n decimal places un­der the pretext t hey will be useful to future compute r scientists or appliedmathemati cian s. Everyone knows t hat t he mathemat icians of the XVIIth andXVIIlth cent ur ies loved numerical comp utations - done by hand, not byt apping the keys of an elect ronic gadget - that ena bled them to verifytheir theoretic al results or to demonstrate the power of t heir methods . T hischildhood sickness of analysis disappeared when in the XIXth century one

Pr eface IX

began addressing rigour of proof and generality of formulations, rather t hanformulae.

This do es not mean that numeri cal calculations have become pointless :thanks to computers, one can do more and more of them, for better or worse,in all scientific and t echnical areas that, from medical im aging to the per­fecting of nuclear weapons'' , use mathem atics. One do es the same in certainbranches of mathem atics too; for example, displaying a large number of curvesmay op en the way to a general theorem or to under st anding a topologicalsit uation, not to speak of the traditional number theory where numericalexpe riment always was, and st ill is, used to formulate or verify conject ures.

This only means that t he aim of an exposit ion of t he prin ciples of anal­ysis is not to teach numerical t echniques. Moreover , t he partisans of appliedmathematics , of numerical analysis and of computer scien ce in all t he uni­versities of t he world manifest their imperialist tende nc ies far too clearly forreal mathematicians to t ake on in their stead a t ask for which they gene rallylack both taste and competen ce.

* * *The innocent reader and many confirmed mathematicians will probabl y besurprised, possibly even shocked, to find in my book some ver y heavy allu­sions to extramat hem at ical subject s and particularly to the relations betweenscience and weaponry. This is neither politically nor scien tifi cally correct : Sci­ence is politically neutral" , even when someone lets it fall inadvert ently onHiroshima while the future winner of a Nobel Prize in physics is recording theresults in a B-29 trailing the Enola Gay7 . Nor is it part of the curriculum: a

5 Nuclear powers agreed a dozen years ago to stop testing. The reason why wasthat testing was made unnecessary by the improvement of numerical analysis.The most immediate consequence of this "progress" is th at everything is nowdone in full secrecy, which was not the case when they had to propel into thest ra tosphere two million tons of radioacti ve rock and sand in order to check their"gadgets" .

6 An assert ion long since demolished by count less stu dies, notably American, ei­ther of part icular facets of scient ific act ivity, or of Science in a general way,e.g. in Bernard Barb er, Science and the Social Order (Collier Books, 1952) andJean-Jacques Salomon , Scie nce et Politiqu e (Paris, Ed. du Seuil, 1970, reed.Economica) . Disclosing th e influence of polit ics, for inst ance of WW II and theCold War, upon Science and Technology is not th e same as going in politics asso many scient ists believe without ever having read any serious historical work.And I do not see why being opposed to the militar y exploitation of mathemat icsand science should be considered as a more political stand th an , for instance,helping Los Alamos or Arzamas to develop their "weapons of genocide" was.

7 In Alvarez: Adventures of a Physicist (Basic Books, 1987), Luis Alvarez's tripto Hiroshim a is t he very first thing he relat es in his book . He was also one ofth e main proponents of the H-bomb and, at th e end of Octob er, 1949, went toWashington to lobby in favor of it . In 1954, he testified at the Oppenheimersecurity hearing that Opp enheimer's opposition to the H-bomb was proof of anexceedingly poor judgment . Alvarez is one of many similar counterexamples tothe "neutrality of Science" theory.

X Preface

scientist's business is to provide his students or readers, without commentary,the knowledge they will later use , for better or for worse, as suits them. Itwill be up to them to discover by themselves, possibly years after graduating,that which "has no place" (why, please?) in scientific books or lectures andwhich was not told them by older scientists well aware of it, or who shouldhave been. Let me give you a few French examples.

As a dozen of people who mostly work in particle physics assured me,you can spend five or eight years learning physics without ever having heardanything about nuclear weapons. I once checked the chemistry library of myParis university for books by Louis Fieser, a most eminent Harvard chemistwho, during WW II, was in charge of improving incendiary weapons anddeveloped napalm; all of his chemistry books are there, but not his accountof war works .

I found another, particularly caricatural, example in a textbook of physicsfor high-school finishing students; as required by the French official instruc­tions in 1995, the concluding chapter, on the laser (never mind that an eigh­teen year old boy or girl can understand practically nothing to it), mentioneda number of civilian applications - ophthalmology, measure of atmosphericpollution, compact discs, energy production by laser-induced thermonuclearfusion", etc. - but not a single military use of lasers, a domain in whichFrench industry was always very strong. This is not only dishonest; it is afoolish way to hide the truth since students read newspapers, look at TV,and if they type "laser military history" on www.google.corn, they will getabout 164,000 documents!

Thirty years ago, in part under the influence of what I had seen on Amer­ican campuses and read in American newspapers and such reviews as Science

8 Louis Fieser, The Scientific Method. A Personal Account of Unusual Projects inWar and in Peace (Reinhold, 1964). In the Biographical Memoirs, v. 65, 1994,of the (American) National Academy of Science, Fieser's biographer has this tosay (p. 165) about his work during WW II : "With the approach of World War II,Fieser was drawn increasingly into war-related projects. A brief excursion into thearea of mixed aliphatic-aromatic polynitro compounds for possible use as exoticexplosives was followed by studies of alkali salts of long chain fatty acids asincendiaries, but by far the most important of his war-related work was his longand intensive study of the quinone antimalarials", to which the author devotesone full page. The word "napalm" is nowhere to be found in this fourteen-pagebiography, a beautiful example of the art of fooling the reader with opaquetechnical jargon. All the more remarkable since Fieser was strongly criticisedduring the Vietnam War for his development of napalm. In his long biographyof von Neumann in the Dictionary of Scientific Biography, J. Dieudonne devotestwo lines to what he calls his "govern ment" work without telling us whetherit had to do with, say, the H-bomb or cancer research , two strongly supporteddomains of "government" work.

9 This is a very long term project, but the French and American military havejustified this very expensive enterprise by pointing out that the new knowledgeof fusion processes it will provide will be used to improve nuclear weapons, a factthat is of course not mentioned in the textbook.

Preface XI

and the Bulletin of the Atomic Scientists, I succeeded, to my great aston­ishment, to convince the head of my Paris universi ty library to start a newsection that would be devoted to what was then called in America Scienceand Society st udies. Although it received lit tle money, you can now find thereseveral thousands of (mostly American) books and the main reviews in thehistory of science and technology, including the military side of it , the armsrace, economics of research and development, science poli cy, etc .: no exclu­sive. But almost all the readers are people who specialise in that field , whil emost of t he 5,000 scienti sts working at the university don 't even know theexistence of this library. Since their specialised librari es are practically emptyin this respect, the conclusion is inescap able: their only sources of informationare their gener ally narrow personal experience!" , perh aps some historical ar­ticl es written in scientifi c reviews by scientists who have no idea of historicalwri ting!", and cafete ria conversations:

The humanist who looks at science from the point of view of his ownendeavours is bound to be impressed , first of all , by its startli ng lack ofinsight into itself. Scient ists seem able to go about their business in a stateof indifference to, if no t ignorance of, any t hing but the going, cur rent lyacceptable doctrines of their several disciplines . . . The only thing wrongwith scientists is that they don 't understand science. T hey don' t knowwh ere their institutions come from, what forces shaped and are st ill sh apingthem , and t hey are wedded to an ant ihist orica l way of thinking whichthreatens to deter them from ever finding out12

.

Itappears more honest to me to violate these miserabl e and far too comfort­abl e taboos and to put on their guard t hose innocents who leap into the darkinto careers of which they know nothing. Because of their past and potential

10 Itis not always that narrow. As in t he USA - the mod el - there are inFrance scien tists who have been for a long time in top govern ment commit teesor who have coope rated with industry. They obv iously know a lot more than t heaverage researcher, let alone stude nt . But they mostly don't speak, much lesswrite, particul arly when defence act ivit ies are involved. This st riking differenc ebetween French and Am erican "Stat esmen of Science" can perhaps be explainedby the fact that the political spec t rum exte nds much far ther to the left in Francet han in the USA, so that defence work was , at least during most of the ColdWar , mu ch mo re cont roversial here t han on the other side of the Atlantic.

11 One of the books I have recently read is Gr egg Herken , Brotherhood of the Bomb:Th e Tangled Lives and Loyalties of Robert Oppenheim er, Ernest Lawrence, andEdward Teller (Henry Holt, 2002), a superb though very concentrated bo ok.The main t ext , 334 pages, is followed by over 2,000 notes: an average of sixreferences to sources per page (and a lot more on Intern et). No active scient istcould spend ten years read ing two hundred bo oks and pap ers already published ,interviewing at length eight y colleagues, discover ing an d reading hundreds ofrecent ly declassified government files , and organizing t his amount of informationinto a coherent book.

12 Eric La rrabee, Science and the Common Reader (Commentar y, June 1966) . As Isai d above, old scient ists who have long been top consultants to their govern mentare not as innocent as Larrab ee puts it , but the new generat ion has not theirexpe rience of science politics .

XII Preface

catast rophic consequences, t he connect ions between science , technology andarmament s concern all who go into science or technology or who practicet hem. T hey have been governed for half a cent ury by the exist ence of publicorgani sations and private enterprises whose function is the system atic trans­formation of scientific and technological progress into military progress withint he limits, often elast ic, of the economic capac it ies of the various count rieswhich t ake part in it :

With the at tent ion wh ich is paid in these days to weapons of war , there isprob ably no known scient ific principl e that has not alr ead y been carefullyscruti nize d to see whether it is of any significance for defence ' :'.

In countries - France is a prime example - where discussions on therelations between Science and Defence have been dominated for decad es firstby silence , t hen by a thick consensusl" , and have been tot ally absent fromuniversity teachlng l", the thing to say to young people is t ha t one of the formsof intellectual liberty is not to let oneself be dominated by the domin ant ideas.

But t his requires access to other informat ion sources . Itwould be impossi­ble to thoroughly discuss this subject and its history within the framework ofa mathematical t reati se. I nevertheless decided to write a few dozen pages ­the Postface to Volume II - in order to give the int erested reader an idea of itand, in par ti cular , to show t hat the question and the subject do exist . I havenot balked at citing a good number of important bibliographical references

13 Sir Solly Zuckerman , Scienti sts and War (London , Ham ish Hamilton , 1962,p . 80) ; the author was at the time the British Government chief scientist an dhad formerly been the head of Br it ish military research . There is no reason tobelieve that Zuckerman 's statement is no longer valid , part icularly in Am er ica .

14 "Science et Defense" is the title of a French associat ion founded in 1983 byCharles Hernu , then t he (socialist ) Secretary of Defence and future hero of theGreenpeace affair - the clumsy sinking in Auckland harbour by French age nts ofa ship t hat would have interfere d with a French nucl ear test in the P acific. Sup­port ed by t he Armam ent branch of Defence, the associat ion organises a yearlycongress, where, over two days, engineers and scienti st s lecture on t he techni­cal problem s of armame nts and t he closely related sciences . Several hundredsof people att end : military, engineers, industrialist s , scient ists , and , inevi t ably,polit ical scient ists an d metaphysicians of strategy. France is, to my knowledge,the only count ry where wh at a number of American historian s now call thescientific-military-industrial complex dares to exhibit itself so publicly and with­out provoking t he least reaction. This would not have been possible before theconversion of the Socialist and Communist parties to nucl ear weapo ns when , atthe end of the 1970s, they saw a good prospect of winning the 1981 presidentialelect ion .

15 Americ a was , in t he 1970s, a no table exception to this general state ment : st u­dent prot ests against t he Vietnam war and t he cooperat ion of many universitydepar t men ts or laboratories with the DoD led some universities to ad d to t he ircurr iculum lectures on various as pects of "Science an d Society" that at t racted asizeable number of science stude nts, while some teachers in the history of sciencesaw their audie nce sudde nly grow. Although t he t rad it ional back to normal pro­cess did no t take very long, many of the pr esent generati on of specialists foundtheir calling during this period .

Preface XIII

- there are plenty more - which will allow t hose who so desire to complet e,verify, or discuss this text. I do not have the naive hop e that a twenty yearold student of mathematics will plunge into this ocean of literature; it wouldha rdly even be a very good service to encourage him to do so. But maybethis text will find readers who are not so young and no longer have to submitto examinations or compet itions for success. Although the French version ofthis Postface devot ed a good deal of space to the French situation, I thoughtit better, in the English version , to emphasise the American sit uation morethan I did in French , and this for several good reasons.

From Pearl Harbor to the present day, America has been t he world leaderin this domain - a leader which, for a dozen year s, has no longer hadany competitor worth naming and seems to be in a technological arms raceagainst its elf?", as was already the case when it spent $2 000 000 000 (onepercent of its 1945 GNP) during WW II in order to get the atomic bomb be­fore t he Nazis, who did not believe it could be available in time and devotedvery little resources to it. This American polarisation on scientifi c weapons,more or less faithfully imitate d in the Soviet Union, Britain and France,had enormous poli ti cal consequences; among others, it compelled the muchweaker Soviet Union to devote to defence a proportion of its resources whichmust have strongly contributed to it s downfall and to the present Americanhegemony. On the other hand, the civilian uses of mostly American militaryinnovations in elect ronics, informatics, aviation, space, telecommunications,nuclear power , etc., had a deep influence on the daily life of people every­where. Wi thout WW II and the arms race, most of these innovations wouldhave come much later , or never , because the finan cing of resear ch, develop­ment and initial production by defence organi sations made it possible forprivate enterprises to t ake risks which, otherwise, would have been barred bythe return on investment principle that governs civilian innovations. WithoutWorld War II , no V-2 missiles and no atomic weapons; without these and theCold War, no int ercontinent al ballistic missiles; without ICBMs and the needof t he central military authorit ies for inst ant worldwide command, cont rol,communicat ion, and inte lligence - C3I as they call it - no satellites; andwithout satellite s and many other innovat ions propelled by the military ­compute rs, integrat ed circuits, Arpanet , etc. - t hen no Internet , to mentiononly this most spectacular spin-off of the arms race. The idea th at civilianindust ry could have, by it self, spent tens or hundreds of billions in order toinvent , produce and market such giganti c amounts of hardware and softwareat a t ime when nobody bu t the milit ary had any proven need for it is foolish .Civilian business does not deal in science fiction .

WW II and the arms race also contributed to propelling the funding ofscientific resear ch proper to levels which, before 1939, would have seemed

16 Ithas been recently disclosed that Am eric a will develop in the next 10 or 15years an hyp ersonic cruise miss ile that will be able to strike anyw here on t heEar th in less than two hours from bases in continental Am er ica.

XIV Pr eface

unrealist ic in t he utmost , a fact of which scientists everywhere were t he firstbeneficiari es alt hough never nearly as much as American ones!" . This not onlymade it possible for many more young Americans to choose scientific careersthan was the case before WW II , it also attracted to America many scient ists(and still more engineers) who had been educated elsewhere, a pro cess t hatis cont inuing to this day - t he famous brain drain that was first notic edin the 1950s, not t o mention some Russian immigrants after 1917 and t heEuropean Jews in t he 1930s.

* **The French version of this book included many citat ions and references inEnglish, par ticularl y in the Postface to Volume II , t his in order to encouragethe read er to use a language that is absolute ly indispensable if one wantsto inform oneself on anything at all: for clear demographic reasons Fran ceaccount s for only a small proportion of the literature, for inst an ce from 3%(technology) to 7% (mathematics) in the sciences on t he world scale, andalt hough French authors publish excellent books in many domains, scient ificor not , t hey cannot be leaders everywhere. T here is for instan ce nothing ofany value on the history of nuclear weapons, not even of French ones, andnon e of t he best American books have been translated. Almost all I know inthe Scienc e and Defence domain has been learned from American authors,alt hough a few French historians of Science and Technology are beginning todeal with it.

There is no need to suggest to readers of the English t ranslation of thepresent book t o learn English . One should however warn the beginner t hat ,even though well over 60% of the mathematical literature is now in English ,an ability to read French is, at t he resear ch level, st ill needed . Since 1945, theFields Medal has been awarded to 44 people worldwide; seven of them wereFrench, and two more, alt hough they are not , did all their previous work inFran ce; the first Abel Prize (a recently created subst itute to the nonexist entNobel Prize for Mathemat ics) has been awarded in 2003 to Jean-Pierre Serre,who won a Fields Medal in 1954, and others won for inst ance t he Wolf Prize.There are in France many more excellent mathematician s than these stars;alt hough some publish in English , still many write in French. And t here areof course German and Russian authors, among ot hers, who st ill publish inthe one language t hey learne d as infants , as anglophone authors always did .

17 In 1965, Isidor Rabi, a winner of the Nobel Pr ize in physics, pointed out that thebudget of the Columbia University physics lab had grown from 15,000 dollars be­fore the war to thr ee millions and attributes this to the war which "did wonderfulthings in some respects" . Hans Bethe, another Nobel Laureate , remembered in1962 that before WW II he found it difficult to get some $3,000 for a cyclotronat Cornell, but that, "Today, $3, 000 is pin m onett. We use it in this laborat oryin a day" _To be objective, one should also note the fantastic increase of civilianresearch funds allocated to Life Sciences, mainly Biology and Medicine; but evenin this case, it was WW II , especially the development of penicillin, which at thestart demonstrated what could be done in these fields with enough money anda concerted effort.

Preface XV

You fortunately don 't need to learn J apanese: Japanese authors do not useit at t he international level, a most courteous stand when you t hink that,for them , learning English is a lo t more work than learning French is for t heAmerican , or English for the French.

T he fact that English has acqui red almost t he status of an internatio na lcommon lan guage, or lingua franca, has of course its upside, and any otherreasonably widespread language, as Lat in was t hree centur ies ago, woulddo . The prevalence of English is often explained by the fact that it is sup­posedly simpler than, say, German , French, or Russian , and that anywayanglophones now form a large proportion of scientists at t he world level.As sugges te d above, this preponderance of English , which goes far beyondScience , is also, and possibly mainly, a corollary of t he enormous resourcesAmerican government, industry and pr ivat e foundati ons have devoted to Sci­ence and Technology since the 1940s and more generally of the overwhelmingsuperiority of t he American econorny l' .

T here is in Fran ce, and pro ba bly elsewhere too, a theory according towhich, t hanks to t he overwhelming power America acquired in 1945 and st illmore in 1990, the result , or even purpose, of the "invasion of English" is tospread across t he whole world the American conceptions of society, politics,economy, technology, mass media, etc. and to help American enterprises toacquire larger and larger parts of foreign markets everyw here, a process t hat,alt hough or because successful, meet s strong op position in many count ries.

Although greatly reinforced by WW II , it started much sooner. T he usein America of such typical exp ressions as "richest in the world" , "greatest inthe world" , "tallest in the world " , "fastest in the world" , "first in the world" ,etc. was already widespread in t he 1900s and was a plain enough symptom.St andard Oil , General Elect ric, Ford were models of multinational compa niest hat European enterprises t ried (generally without much success at t he time,if you except I.G. Farben in t he 1920s) to imit ate. American sewing machines,typ ewrit ers and acco unt ing machines, agr icultural machinery, machine toolsan d, between WW I and WW II , automobiles were invading Europe longbefore computers did. In t he 1920s, jazz had already its fan ati cs everywhere,Hollywood 's movies had already 60-80% of t he French market , most of thebest movie t heaters were in American hands, and t he answer to French at­te mpts t o impose imp ort quotas was a near total boycott of French moviesin America (19 in 1929, against hundreds of American movies in France) ,a situatio n which did not improve after WW II. After 1918 it was Wilson ,a U.S. P resident with the mind and eloquence of a Protest ant missionary,who launched the Society of Nations, which Congress rejected . The UnitedStates' intervent ionist policy was already quite plain in t he Americas, Chinaan d J ap an long before t he end of t he XIX th cent ury, and as a recent book -?

18 In some French companies , meeti ngs of t he Board are in English because of t hepresence of one or two American members. At t he present t ime, about 20% oft he total capit alisat ion of t he Paris Stock Exchange is American-owned.

19 P hilippe Roger, L 'ennemi americain. Cenealoqie de l'anticmericamisme fran cais(Paris, Seuil, 2002, 600 pp.) puts everyt hing in historical perspect ive wit hout

XVI Preface

reminded us, French hostility towar d America was powerfully increased bythe war against Spain in 1898, which was viewed by the French right as athreat to European colonial empires, and by t he left as a conclusive proof oft he t ransformat ion of an already unpalat abl e American capitalism int o out­right imperi alism or economic colonialism . As to the present American t ast efor firearms, a unique feat ure am ong "civilised" count ries as they were calledin the 1900s, it was Samuel Colt who , during the war aga inst Mexico, trig­gered the craze by adopting the American syste m of manufacture invente din arsenals in order to mass produce his celebrate d revolvers. Present in­equalit ies in the distribution of income are not worse than t hey were at thet ime John D. Rockefeller was worth over one billion dollars, i.e. ab out 2% ofAmerica 's GNP: a propor ti on which, nowadays, would amount to some 200billion . And New York bar owners pouring French wine on t he st reet 20 wereseen already long before Mar ch 2003.

Thus, nothing very new under the sun, except that American int ern a­tional preponderance and unil ateralisrn have now acquired the status of anofficial doctrine supporte d by a host of ideologist s invoking a fundamental­ist Protest ant ethic in order to justify interventions which, in the eyes of avast majority of peo ple everywhere, are nothing bu t displ ays of power evenwhen they rid nations of barbar ic rul ers or religious oppression in the hopeof establishing t here a (probabl y very weak) version of Western democracy.

T hat being said , nobody has to appreciate the barbaric music and violentmovies which presently come from America (the American stars of my yout hwere Charlie Ch aplin , Buster Keaton and the Marx Brothers) . Americans donot merely dict at e the export of these pro du ctions through int ern ational com­mercial agreements and by owning big dist ribution companies; they also sellthem by findin g ind igeno us customers (or imitators) who are only t oo happyto make money by distributing them among a young and most often uncul­tured public. And how would local television fill it s hours of programmes,how could the cinemas function, without the flow of American productions?The work force in France (say) is not large enough to replace American medi­ocrity with French medio crity; and no count ry is capable of producing a newShakespeare or a new Bar t ok every day. One therefore broadcast s what isavailable or imitates American crass "games" .

himself falling into the trap. Itgoes without saying that many of the criticismsthat some French intellectuals and politicians of the right addressed to Americawould apply just as well to France. Howard Zinn, A People's History of the UnitedStates, 1492-Present (HarperCollins, 2003 edit ion) , while or because very one­sided, would be very useful to help understand criticism from the French left ,which was never as systematic and well organised as Zinn's, not to mention booksby Lewis Mumford , Noam Chomsky, etc .

20 Ifa boycott of French wines were to lower prices in France, I, for one, wouldnot shed crocodile tears at the tragic fate of poor American patr iots heroicallydepriving themselves of Chateau Latour at $1,000 a bottle (assuming they don'thave a stock of it in their cellar).

Preface XVII

Nor is one obliged to approve t he Darwinian concept s of economic com­petition and social relations which, thanks to technologies t hat have emergedstraight from the cold war and arms race, are presently expa nding und erthe name of "globalisat ion" : the exte nsion to the planet of a "liberal", i.e,capitalist, and "modern" economic system founded on the principles isolat edby Adam Smith in 1776 and assimilate d err oneously by the robber baronswho , at the end of the XIX t h century, erected the great American capitalistente rprises, aft erwards revised a little and codified . It is now forbidden toshoot the strikers but not to dom esticate the unions; to dismiss thousandsof employees to improve the competitiveness of companies and in return toexploit the work force at low pay in developing countries; to push for thedism antl ement of European social welfare systems hard won afte r a centuryof struggle but now judged t oo expensive - or smackin g of Socialism ? - bythe alumni of the Harvard Business School an d its foreign imitations; to sub­orn the public market s by handing cheques to polit ical parties as is presentlythe case in Fran ce, Germany, Italy, etc ., or, in the Third World, to gangstersin high places in order to inundat e t he count ries they rule or own with killingmachines under the pretext of lowering the un it price for the count ries thatproduce them 21 , or in order to secure the rights to exploit ing their naturalresources. It is the reign of money, whose rallying slogan was launched a hun­dred and fifty years ago by a famous French minister: Enrichissez-vous! Ifyou can22 ...

That said , America possesses, notably in its universities, an int ellectualclass not to be globally confused with the spokesmen of the Pentagon 's war­lords or the operators of Wall Street. In particular and as I said above, no one,in Fr anc e, has revealed the military influence on scientific and technologicaldevelopment since 1940 as a number of American historians, particularly ofthe younger generation, have don e for a quarter of a cent ury with the helpof massive do cumentation; if you are interest ed in , say, the history of theCold War, you will find in the American literature all the information, pointsof view and opinions you want. There is no need eit her to point out thatmany Americ an novelist s did not wait until 2003 to disseminate unortho­dox descriptions of the American society. As to the mathematicians, manyof whom have always been very crit ical of official policy, the year s I spentwith my famil y in the 1950s and 1960s at Urbana , Berkeley and Princetonwere among the happiest of my life. And when, at t he end of October 1961,

21 A few years ago , the sale to Taiwan by the French company T homson-C SF ofvery sophist icated frigat es generated a $500 million return for (mostly unknown)Taiwanese and French politi cian s or political parties and go-betweens . An inves­tigation of t he case by the French judiciary was sto pped in a most elegant way:t he Dep ar tment of Defen ce classified all T homson-C SF documents perta ining toit. The company decid ed a few months later to change its name into T hales , arat he r unpalat able reference to Mathem atics.

22 What Guizot said is : Get rich , by your work and savings - a cynical precept ata time wh en the overwh elming majori ty of people , afte r working twelve hours aday six days per week , would die as po or as their parents were.

XVIII Preface

my Pari s flat was destroyed'< because I had , rather mildly in fact , spokenout during a lecture against the savage repression in Paris of a peaceful Al­gerian demonst ration for independ ence, I received two day s lat er a telegramfrom J . Rob ert Oppenheimer inviting me, on very generous te rms, to spendthe remainder of the academic year at t he Princeton Insti tute; we went twomonths later , and there my wife recovered her usual balance.

It goes without saying t hat t he facts and opinions to be found in this bookare my own and full responsibility. They do not commi t Springer- Verlag toany degree. Some will perhaps reproach my publisher for not having censore dme. Being ill-placed to do so in their place I prefer , for myself, to thankthem warmly for having been allowed t he liberty to express myself. This isan at t it ude which I sure ly would not have encountered everywhere and whichI appreciate for its proper worth.

Paris 2003.Rgodement@aol.com

23 My wife, who was there, was ext remely lucky not to be hurt , though she wasbadly shaken for months afterwards; my three child ren came home from schoolfift een minutes a fte r t he bombing. T he very first thing I noticed after climbing aladder to my place was t he police searching my papers (there was nothing to befoun d) . T hough they certainly ident ified t he authors of this a t tempt - prob ablycandidates to t he French equivalent of West Po int schoo l, acco rding to my ownstude nts - I was never told anything and I very mu ch doubt that t hey ever hadto bear any unpleasan t consequences.

Contents

Preface . . . . . . . . . .. .. . . . . . . . . .. . . . . . . .. . . . .. . . . . . . . . . . . . . .. . . . . V

I - Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1§l. Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 - Membership, equality, empty set . . . . . . . . . . . . . . . . . . . . . . 72 - The set defined by a relation. Intersections and uni ons . . . 103 - Whole numbers. Infinite sets. . . . . . . . . . . . . . . . . . . . . . . . . . 134 - Ordered pair s, Cartesian products , sets of subsets . . . . . .. 175 - Functions, maps, corres pondences. . . . . . . . . . . . . . . . . . . . . 196 - Injections, surjec t ions , bijections . . . . . . . . . . . . . . . . . . . . . . 237 - Equipotent sets. Count able sets. . . . . . . . . . . . . . . . . . . . . .. 258 - T he different typ es of infinity . . . . . . . . . . . . . . . . . . . . . . . . . 289 - Ordinals and cardinals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

§2. T he logic of logicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

II - Convergence: Discret e variables . . . . . . . . . . . . . . . . . . . . . . . . .. 45§l. Convergent sequences and series . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

o- Introd uction: what is a real number? 451 - Algebraic operations and t he order relation: axioms of JR . 532 - Inequalities and intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 - Local or asymptot ic properties . . . . . . . . . . . . . . . . . . . . . . .. 594 - T he concept of limit . Continuity and differentiability . . . . 635 - Convergent sequences: definition and examples . . . . . . . . . . 676 - T he language of series . . . . . . . . . . . . . . . . . . . . 767 - The marvels of t he harmonic series . . . . . . . . . . . . . . . . . . .. 818 - Algebraic operations on limit s ... . . . . . . . . . . . . . . . . . . . . . 95

§2. Absolutely convergent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 989 - Increasing sequences. Upper bound of a set of real numbers 9810 - The function log x . Roots of a posit ive number 10311 - What is an integral? 11012 - Series with positive terms 11413 - Alt ernating series 11914 - Classical absolute ly convergent series 12315 - Unconditional convergence: genera l case 127

XX Conte nt s

16 - Comparison relations. Criteria of Cauchy and d 'Alemb ert 13217 - Infinite limits 13818 - Unconditional convergence: associativity 139

§3. First concept s of analyt ic functions 14819 - The Taylor series 14820 - The principle of analyt ic cont inuat ion 15821 - T he function cot x and the series :Ll /n2k 16222 - Multiplication of series. Composition of analyt ic func-

tions. Formal series 16723 - The ellipt ic functions of Weierstrass 178

III - Convergence: Continuous variables 187§l. The intermediate value theorem 187

1 - Limit values of a function. Op en and closed set s 1872 - Continuous functions 1923 - Right and left limi ts of a monotone function 1974 - The intermediate valu e theorem 200

§2. Uniform convergence 2055 - Limits of cont inuous functions . . . . . . . . . . . . . . . . . . . . . . . . 2056 - A slip up of Cauchy's 2117 - The uniform metric 2168 - Ser ies of cont inuous functions. Normal convergence 220

§3. Bolzano- Weierstrass and Cauchy's criterion 2259 - Nested int ervals, Bolzano-Weierstrass, compac t sets 22510 - Cauchy's general convergence criterion 22811 - Cauchy's criterion for series: examples 23412 - Limits of limits 23913 - Passing to the limit in a series of functions 241

§4. Differentiabl e functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24414 - Derivatives of a function 24415 - Rules for calculating derivatives 25216 - The mean value theorem 26017 - Sequences and series of differenti abl e functions 26518 - Extensions to unconditional convergence 270

§5. Differentiabl e functions of several vari abl es 27319 - Partial derivatives and differentials 27320 - Differentiability of functions of class C 1 27621 - Differentiation of composite functions 27922 - Limits of differentiable functions 28423 - Interchanging the order of differentiation . . . . . . . . . . . . . . 28724 - Implicit functions 290

Contents XXI

Appendix to Chapter III 3031 - Cartesian spaces and general metric spaces . . . . . . . . . . . . . 3032 - Op en and closed sets 3063 - Limits and Cauchy 's crite rion in a metric space; complete

spaces 3084 - Continuous functions 3115 - Absolutely convergent series in a Ban ach space 3136 - Continuous linear maps 3167 - Compact spaces 3208 - Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

IV Powers, Exponentials, Logarithms, Trigonometric Func-tions 325§1. Direct construction 325

1 - Rational exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3252 - Definition of real powers 3273 - T he calculus of real exponents 3304 - Logarithms to base a. Power fun ctions . . . . . . . . . . . . . . . . . 3325 - Asympto tic behaviour 3336 - Char acterisations of the exponent ial, power and logarith-

mic funct ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3367 - Derivatives of the exponent ial funct ions: di rect method . . 3398 - Derivat ives of exponent ial functions, powers and logarithms342

§2. Series expansions 3459 - The number e. Napierian logarithms 34510 - Exponential and logarithmic series: direct method . . . . . . 34611 - Newton 's binomial series 35112 - The power series for the logarithm 35913 - The exponential fun ction as a limit 36814 - Imaginary exponent ials and t rigonometric functions 37215 - Euler's relati on chez E uler 38316 - Hyperbolic functions 388

§3. Infinite product s 39417 - Absolutely convergent infinite products 39418 - The infinite product for the sine function 39719 - Expansion of an infinite product in series 40320 - Strange identities 407

§4. The topology of the functions A rg(z) and LOg z 414

Index , 425