foundations of real and abstract analysis - axler , gehring , ribet

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1、FoundationsofRealandAbstractAnalysisDouglasS.BridgesSpringerDedicatedtothememoryofmyparents:DouglasMcDonaldBridgesandAllisonHoggSweetAnalytics,’tisthouhastravishedme.Faustus(Marlowe)Thestonewhichthebuildersrefusedisbecometheheadstoneofthecorner.Psalmcxviii,22....fromsosimpleabeginn

2、ingendlessformsmostbeautifulandmostwonderfulhavebeen,andarebeing,evolved.Theoriginofspecies(Darwin)PrefaceThecoreofthisbook,Chapters3through5,presentsacourseonmetric,normed,andHilbertspacesatthesenior/graduatelevel.Themotivationforeachofthesechaptersisthegeneralisationofaparticular

3、attributeofthenEuclideanspaceR:inChapter3,thatattributeisdistance;inChapter4,length;andinChapter5,innerproduct.Inadditiontothestandardtopicsthat,arguably,shouldformpartofthearmouryofanygraduatestudentinmathematics,physics,mathematicaleconomics,theoreticalstatistics,...,thispartofth

4、ebookcontainsmanyresultsandexercisesthatareseldomfoundintextsonanalysisatthislevel.ExamplesofthelatterareWong’sTheorem(3.3.12)showingthattheLebesguecoveringpropertyisequivalenttotheuniformcontinuityproperty,andMotzkin’sresult(5.2.2)thatanonemptyclosedsubsetofEuclideanspacehastheuni

5、queclosestpointpropertyifandonlyifitisconvex.Thesadrealitytodayisthat,perceivingthemasoneoftheharderpartsoftheirmathematicalstudies,studentscontrivetoavoidanalysiscoursesatalmostanycost,inparticularthatoftheirowneducationalandtechnicaldeprivation.Manyuniversitieshaveattimescapitula

6、tedtothenegativedemandofstudentsforanalysiscoursesandhaveseriouslywatereddowntheirexpectationsofstudentsinthatarea.Asaresult,mathematicsma-jorsaregraduating,sometimeswithhighhonours,withlittleexposuretoanythingbutarudimentarycourseortwoonrealandcomplexanalysis,oftenwithoutevenanint

7、roductiontotheLebesgueintegral.Forthatreason,andalsoinordertoprovideareferenceformaterialthatisusedinlaterchapters,Ichosetobeginthisbookwithalongchapterprovidingafast–pacedcourseofrealanalysis,coveringconver-xPrefacegenceofsequencesandseries,continuity,differentiability,and(Riemanna

8、ndRiemann–Stieltjes)integr