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1、泛函分析期末综合实验实验课程名称泛函分析综合实验学院数学与统计学院年级2009专业班数学与应用数学学生姓名王磊学号20092237开课时间2011至2012学年第2学期ContentsContents1MathematicalConcepts11．Q1&2(LinearAnalysis)12．Q3(Topology)2Mathematicsinpapers3MathematicalinApplication4MathematicalConceptsWhenwerefertoMathematical
2、Concepts,wemeanthatthephrasesorthewords,whichconsistofspecialsymbols,describeamathematicalideaprecisely.Mathematicalconceptscanbesubstitutedinothermathematicalexpressionsbyinduction.Anyway,whatconnectsdifferentideasisMathematicianThinking,whichwillbed
3、iscussedlater.Hereareanexamples1．Q1&2(LinearAnalysis)a)IfXisaninfinite-dimensionalnormedvectorspacethenneitherD={x∈X:1}norK={x∈X:=1}iscompact.i.Analysis:ItdoesnotmatterwhetherweuseDorKtoprovethetheorem.Ifthehypothesesholdforthefirstone,thensodothemfort
4、hesecondone.ThestructurepropertiesofDplayedintheproofofthetheoremarethatwheneverwechoosethembyRiesz’Lemma,pointsareofthesameproperties,forexample,hencebeingasequence,andifthesetisotherwiseopen,theproofwillnotbeefficientasinfinite-dimensionalnormedvect
5、orspace,aclosedandboundedsetisequivalenttoacompactset.ii.Examples:Wewillillustratedtheessencescontainedintheproofandcrucialimportantreasonwhywechoosecompactasanidentifieroffiniteorinfinite.1.Xisaninfinite-dimensionalnormedvectorspacesuchthat={x∈X:1}isc
6、ompact.Analysis:First,ifXisafinite-dimensionalspace,thatasetSiscompactisequivalenttoSwhichisaboundandclosedset.Second,willnotbecompactifXisafinite-dimensionalspace.ButwhatifXisainfinitespace?Theconclusionisderivedsincethepropertiesoncompactoftwokindso
7、fspaceareoppositetoeachother.2.AmethodtoconstructwhichcanmaintainpointsselectedbyRiesz’Lemmatobedroppedintothroughlineartransformation.3.LetXbeanormedspace,andlet.IfY,asubspaceofX,isopen,thenY=X.Proof:YisasubspaceofXYisopeninXHowever,if,,whichisacontr
8、adiction!Asaresult,weconcludethatY=X.b)null1．Q3(Topology)a)Definition:IfXisaspaceandAisasetandifisasurjectivemap,thenthereexistsexactlyonetopologyonArelativetowhichisaquotientmap;itiscalledthequotienttopologyinducedby.i.Analysis:Atopologyassoc
