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《Springer.Bridges.Foundations.Of.Real.And.Abstract.Analysis.(Gtm.174.Springer.1998)》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、Dedicatedtothememoryofmyparents:DouglasMcDonaldBridgesandAllisonHoggSweetAnalytics,’tisthouhastravishedme.Faustus(Marlowe)Thestonewhichthebuildersrefusedisbecometheheadstoneofthecorner.Psalmcxviii,22....fromsosimpleabeginningendlessformsmostbeautifulandmostwonderfulhavebeen,andarebeing,ev
2、olved.Theoriginofspecies(Darwin)PrefaceThecoreofthisbook,Chapters3through5,presentsacourseonmetric,normed,andHilbertspacesatthesenior/graduatelevel.ThemotivationforeachofthesechaptersisthegeneralisationofaparticularattributeofthenEuclideanspaceR:inChapter3,thatattributeisdistance;inChapte
3、r4,length;andinChapter5,innerproduct.Inadditiontothestandardtopicsthat,arguably,shouldformpartofthearmouryofanygraduatestudentinmathematics,physics,mathematicaleconomics,theoreticalstatistics,...,thispartofthebookcontainsmanyresultsandexercisesthatareseldomfoundintextsonanalysisatthisleve
4、l.ExamplesofthelatterareWong’sTheorem(3.3.12)showingthattheLebesguecoveringpropertyisequivalenttotheuniformcontinuityproperty,andMotzkin’sresult(5.2.2)thatanonemptyclosedsubsetofEuclideanspacehastheuniqueclosestpointpropertyifandonlyifitisconvex.Thesadrealitytodayisthat,perceivingthemason
5、eoftheharderpartsoftheirmathematicalstudies,studentscontrivetoavoidanalysiscoursesatalmostanycost,inparticularthatoftheirowneducationalandtechnicaldeprivation.Manyuniversitieshaveattimescapitulatedtothenegativedemandofstudentsforanalysiscoursesandhaveseriouslywatereddowntheirexpectationso
6、fstudentsinthatarea.Asaresult,mathematicsma-jorsaregraduating,sometimeswithhighhonours,withlittleexposuretoanythingbutarudimentarycourseortwoonrealandcomplexanalysis,oftenwithoutevenanintroductiontotheLebesgueintegral.Forthatreason,andalsoinordertoprovideareferenceformaterialthatisusedinl
7、aterchapters,Ichosetobeginthisbookwithalongchapterprovidingafast–pacedcourseofrealanalysis,coveringconver-xPrefacegenceofsequencesandseries,continuity,differentiability,and(RiemannandRiemann–Stieltjes)integration.Theinclusionofthatchaptermeansthattheprerequisiteforth
