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1、TimothyGowersMATHEMATleSAVeryShortIntroductionOXFORDUNIVERSITYPRESSOXFORDUNIVERSITYPRESSGreatClarendonStreet,Oxfordox26npOxfordUniversityPressisadepartmentoftheUniversityofOxford.ItfurtherstheUniversity'sobjectiveofe_lIenceinresearch,scholarship,andeducati
2、onbypublishingworldwideinOxfordNewYorkAucklandBangkokBuenosAiresCapeTownChennaiDaresSalaamDelhiHongKongIstanbulKarachiKolkataKualaLumpurMadridMelbourneMexicoCityMumhaiNairobiSaDPauloShanghaiSingaporeTaipeiTokyoTorontov.ithanassociatedcompanyinBerlinOxfordi
3、saregisteredtrademarkofOxfordUniversityPressintheUKandincertainothercountrîesPublishedintheUnitedStatesbyOxfordUniversityPressInc.,NewYork©TimothyGowers2002ThemoralrightsoftheauthorhavebeenassertedDatabaserightOxfordUniversityPress(makerlFirstpublishedasaV
4、eryShortIntroduction2002Ailrightsreserved,Nopartofthispublicationrnaybereproduced,storedinaretrievalsystem~ortransmitted,inanyformorbyanymeans,withoutthepriarpermissioninwritingofOxfordUniversityPress,orasexpresslypermittedbylaw,orundertermsagreedwtththeap
5、propriatereprographiesrightsorganizations.Enquiriesconmingreproductionoutsidethe$Copeoftheabove,houldbesenttatheRightsDepartment,OxfordUniversityPress,attheaddressaboveYoumustnotcirculatethisbookinanyotherbindingDrcoverandyoumustimposethisSaIne{conditionon
6、anyacquirerBritishLibraryCataloguinginPublicationDataDataavailableLibraryofCongressCataloginginPublicationDataDataavaiJableISBN978-0-19-285361-514161820191715'IYPesetbyRefineCatchLld,Bungay,SuffolkPrintedinGreatBritainbyAsbfordColourPressLld,Gosport,Hampsh
7、ireContentsPrefaceixListofdiagramsxiii1Models12Numbersandabstraction173Proofs354Limitsandinfinity565Dimension706Geometry867Estimatesandapproximations1128Sornefrequentlyaskedquestions126Furtherreading139Index141PrefaceEarlyinthe20thcentury,thegreatmathemati
8、cianDavidHilbertnoticedthatanumberofimportantmathematicaJargumentswerestructurallysimilar.Infact,herealizedthatatanappropriatelevelofgeneralitytheycouldberegardedasthesame.Thisobservation,andotherslikeit,gave
