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An Introduction to Nonstandard Real Analysis (Pure and Applied Mathematics (Academic Press), Volume 118) PDF

247 Pages·1985·9.44 MB·English
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Preview An Introduction to Nonstandard Real Analysis (Pure and Applied Mathematics (Academic Press), Volume 118)

AnI ntrodutcot ion NonstanRdeaaArlnd a lysis Thiiss a volumie n PURE AND APPLIMEADT HEMATICS A SerieMso noofg rapahnsdT extbooks EditoSrAs:M UEELl LENBAENRDHG Y MABNA SS Al i�t otfi trilentec hsesi nestr aipepesa artst heen odf t hviosl ume. AnI ntrodutcot ion Nontndard s a RelaA nlaysis ALBERT E. HURD DepartmoefnM ta thematics UniverosfiV tiyc toria VictorBirai.t iCsohl umbia Canada PETER A. LOEB DepartmoefnM ta thematics UnviersoiftyI llinois Urbana1,/ linois 1985 ACADEMIC PRESS, INC. (HarcoBurratc Jeo vanovPiuchb,l ishers) OrlandSoa nD iegoN ew York London TorontMoo ntreaSly dneyT okyo COPYRIGHTIIC>985B YA CADEMIPCR ESSI,N C. ALLR IGHTRSE SERVED. NOP ARTO FT HIPSU BLICATIONB EMR AEYP RODUCOERD TRANSMITTEIDN ANFYO RMO RB YA NYM EANSE.L ECTRONIC ORM ECHANICAILN.C LUDIPNHGO TOCOPYR.E CORDINOGR,A NY INFORMATIOSNTO RAGEA NDR ETRIEVSAYLS TEWMI.T HOUT PERMISSIIONWN R ITINFGR OMT HEP UBLISRH.E ACADEMPIRCE SISN,C . Orlando, Florida 32887 UnitKeidn gdom Editiobny published ACADEMPIRCE SISN C(.L ONDOLNT)D . 24/O2v8a Rloa dL,o ndNoWn I 7DX LibroaryfC ongress CianPt ualbolgiiDncagat tai on Maienn tryun dert itle: An introducnt tioo nonstandarreda la nalysis. Includbeisb liographircefaelr enceasn di ndex. 1.a thMematicala nalysisN,o nstandar1d,.H urd,A . E. (AlberEtm ers)o,n DATE • II.o ebL,P . A. QA299.82.115988 5 515 84-24563 ISB0N- 12-36244(0a-11k .p aper) PRINTIEND TUHNEI TSETDA TI::.S OF AME:RICA 85 86 8887 987654321 Dedicattoe dmt ehmeo royf ABRAHARMO BINSON ThiPsa geI ntentioLneaflBtll ya nk Contents Prefa.c e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . ... . . . .. . . . .. . . . . . . . . . . 1x ChaptIe r lnfiniteasnidmT ahleCs a lculus I.I TheH yperreaNlu mberS ysteamsa nU ltrapowe..r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 •-TransfoofrmR se lati.o..n..s... ....................................................... 8 1.3 Simple LangfuoarRg eelsa tional S.y s.t .e m.s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II 1.4 InterpretaotfiS oinm plSee ntenc...e.s.. .............................................. 15 1.5 TheT ransfPerri ncifpolrSe i mplSee ntenc..e..s.. .................................. 19 1.6 InfiniNtuem bersl,n linitesiamnadtl hse,S tandaPradrtM ap. ....................2 4 I7. TheH yperinteg. .e r.s . . . . . . . . . . . . . . . .2 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 SequencaensdS erie..s.. ............................................................... 32 1.9 Topologoyn t heR eal.s.. ............................................................... 39 I.1 0 Limiatnsd C ontinu.i...t.y... ........................................................... 44 II.I Differentia. . t. i. .o ...n .. . ... . . . . . . . . . ... .. . . . . . ... ... . . .. . .. .. ... . . . ... ... .. ... .. . .. ... . . 51 1.12R iemanInn tegiroan.t ...................................................................5 6 I1.3 SequencoefFs u ncti.o..n..s. ........................................................... 60 1.14 Two ApplicattiooD nisf fereEqntuiaatli o.n..s.. .................................... 63 1.15P rooofft heTr ansfPreirn cip..l..e... ................................................. 67 ChaptIeIr Nonstandard AonnaS luypseirss trreusc tu 111. Superstcrutur.e.s.. ....................................................................... 71 11.2L anguagaensdI nterpretatfioorSn u perstruct..u..re..s... .......................... 74 113. Monomorphisbemtsw eenS uperstruurcetsT:h eT ransfPerirn cip.l..e.. ......... 78 11.4T heU ltrapoCwoenrs tructfioorSn u perstructu..r..e..s.. .......................... 83 11.5H yperliniSteet sE,n largemeanntdCs o,n curreRnetl ati.o..n.s.. .................8 8 116. Intelra nnadE xternaEln titiCeomsp;r ehensiv.e..n..e...s..s.. ..................... 94 11.7T heP ermanence Pr.i..n...c.i...p...l.e.. ............................................ 100 11.8K -SaturSautpeerds tructu. .re.s . .. . . . . . . .. . .. .. . . . .. . . ... .. .. .. ... .. . . . . . .. .. . .. .. . . . . 104 vii viii Contents ChaptIlel r NonstandTahredo royfT opologSipcaacle s III.I BasiDce finitainodRn ess ul.t s. . . . . . . . . . . . . . 1.1 .0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.C2o mpactsnse . . . . . . . . . . . . . . . . . . . . 1.2 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II3I .M etircS pace.s . . . . . . . . . . . . . . . . . . .1 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II4I .NormeVde ctoSrp acsea ndB anachS pace.s . . . . . . . . . . .1 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 111.I5n ner-PrtoS dpuacceasn dH ibe rt Space.s ... ..... ... . .. ..... . ..... . ...... .... 1.4 .5 .. .. 111.N6o nstandHaurldlo sf M etric Sp.a c.e s. . . . . . . . . . .1 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.7 I Compactifica. .t i.o n.s . . . . . . . . . . . . . . . 1.5 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.F8u nctiSopna ce.s.. ................................................................... . 160 ChaptIeVr NonstandaIrndt egraTthieoonr y IV.I StandardizoaftI inotnesrni anlt egraSttirounc tu..r ..e. s . .... . . .. ... . . . .1..6 5.. .. .. . . IV. 2 MeasurTeh eorfyo rC ompleItnet egraSttiruocnt ur. .e s. . . . . . . 1.7 5. . . . . . . . . . . . . . . . . . IV. 3I ntgreatioonnR "t;h eR iesRze presentTahteioorne ..m . . . . ..... .. 1..8 9.. . .. . . . . . . . . IV. 4 Basic Convergence. .T h.e o.r e.m s. . .. . .. .. . . 1. 95. . . . . . . . .. .. .. . . . . . . . . . . . . . . . . . . . . . . .. IV.5 TheF ubini The.o.r.e.m. .......................................................2.00. .... IV.6A pplicattioSo tnso chasPtrioccs eses.. .............................................2 05 Appendix Ultrafi..l...t..e..r..s... ............................................................... 219 References 222 List of .S...y....m..b...o...l...s... .... ..................................................... 225 Inde..x.. .................................................... ...... ............................... 227 Preface Then otioofna ni nfinitehsaisam papela reodf fa ndo ni nm athematsiicnsct eh e timoef A rchimedIensh .i sf ormulaotifto hnec alcuilnut sh e1 670tsh,eG erman mathematiWciilahne lGmo ttfrLieeidb ntierza teidn finitesiamsia dlesa nlu m­ bersr,a thleirk iem aginanruym berwsh,i cwhe res mallienar b solute value than anyo rdinarreyan lu mberb utw hicnhe vertheolbeesysea dl olf t heu sual loafw s arithmeLteiicb.n riezg ardiendf initesaisam uaslesf fuilc tiwohni cfha cilitated mathematciocmaplu tatainodin nv entiAoln.t houigtgh a inerda piadc ceptaonnc e thec ontinoefnE tu ropeL,e ibnimze'tsh owda sn otw ithoiuttds e tractIonr s. commentionngt hef oundatioofcn asl culasud se velopebdo thb yL eibnainzd NewtonB,i shoGpe orgBee rkelweryo te",A ndw hata ret hessea mee vanesncte incremenTthesy?a ren eithfeirn iqtuea ntitnioerqs u,a ntiitnifiensi tsemlayl nlo,r yetn othinMga.y w e notc altlh emt heg hosotfsd eparteqdu antitiTehseq ?u"e s­ tiowna s,H ow cant herbee ap ositniuvmeb ewrh icihs s malltehra ann yr eal numbewri thobueti nzge roD?e spittheiu sn answerqeude stitohnei, n finitesimal calcuwlaussd evelopbeyEd u learn do thedrusr nigt hee ightntehea ndn ineteenth centuriinetason i mpressbiovdeyo fw orkI.tw asn otu nttihlle a tnei netntehec en­ turtyh aatna dequate deoffil niimtriietop nl actehdce a lcuolfui sn finitesainmda ls providaer di gorofouusn datfiooarnn alysis. Following thtihseu sdee velopment, ofi nfinitesgirmaadlusa flaldye dp,e rsistoinnlgya sa n intuitaiivdte o c on­ ceptualization. Therteh e matsttoeorud n ti1l9 60w henA brahaRmo binsgoanv ea rigorous foundatfioortn h eu seo fi nfinitesinia mnaallsy sMiosr.es pecificaRlolbyi,n son showetdh atth es eotf r eanlu mbercsa nb er egardaesad s ubsoefta largseerot f "number(sc"a llheydp errenaulm berwsh)i ccho ntaiinnsf initesainmdaa lloss, witahp propridaetfeilnyae rtdi thmeotpiecr atisoantsi,s afiloelfs t hea rithmetic ruleosb eyebdy t heo rdinarreyan lu mberEsv.e nm oreh,e d emonstrtahtaettdh e relatisotnraulc tovuerret her eal(ss etrse,l atieotncsc.,a) n b ee xtendteoad sim­ ilasrt ructouvreetr h eh yperreiansl usc ahw ayt haatl slt atemetnruteis n t her eal structruermea itnr uew,i tah suitaibnltee rpretaitnti hoenhy ,pe rreals tructure. Thilsa ttperro pertkyn,o wnas t hter anspfreirn ciipstl hepe,i votraels uolfRt o bin­ son'dsi scovery. ix

Description:
The aim of this book is to make Robinson's discovery, and some of the subsequent research, available to students with a background in undergraduate mathematics. In its various forms, the manuscript was used by the second author in several graduate courses at the University of Illinois at Urbana-Cham
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