实分析与复分析作业节选

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1、实分析与复分析I,实分析作业1.设打是线性空间X的起平面.4二且是X的仿射集。试证軽'或者山比或者A=X・2.设•卫为说个线性空间.4K-F为线性映射，Au兀Buy为两个空间中的凸集.试证明：-J/4)n{yW丫曰为三九4r=對};疋7(月)=归丘区

2、划££,21制=期}禄是凸集.3.CallasetHuR"ahyperplaneifthereexistrealnumbersan・.・,an,c(witha：H0foratleastonei)suchthatHconsistsofallpointsx=

3、(xnx„)thatsatisfy工厲冲=c.Suppose£isaconvexsetinR",withnonemptyinterior,andyisaboundarypointofE.ProvethatthereisahyperplaneHsuchthatyeHandEliesentirelyononesideofH・4.LetXbeareallinearspace,AaconvexsubsetofX,andVanaffinesubspaceofX.WesaythatVsupports4ifXnTi

4、sanon-emptyextremalsubsetofA.(Thisreducestothedefinitionin6CwhenVisahyperplane.)a)Let£beaconvex/-extremalset.Showthattheaffinesubspaceaff(E)supports4b)IfXispartiallyorderedwithpositivewedgeP,asubspaceMuXisanorderidealiftheorderinterval{x:yWxWz}liesinMwh

5、enevery.zeM・ShowthatMisanorderidealifandonlyifMsupportsP.5.试证胜如杲川』为X中的两个不相交的凸集，那么存在两个互补凸集£卫使得4U0EuQ〈角谷静夫-Stoue定理人6.Sometimesitisofinteresttoknowwhentwodisjointconvexsetscanbe(strongly)separatedbyagivenhyperplane.Thesimplestcaseisthefollowing:wearegivenp

6、ointsa,b,pl9P2,・・・，4inRnwiththeingeneralposition.LetHbethehyperplaneaff({p!,・・・，pM}).AssumingthatneitheranorbliesinH,showthatHstronglyseparatesaandbifandonlyifthedeterminantsdet(N・・・,瓦)anddet(K,0“・・・,瓦)haveoppositesigns,whereforx=(Q,・・・，x=thecolumnvecto

7、r(&,・・・,§“，1)T.(Considertheconditionforthelinesegment(a,b)tointersectH.)7.Let/beaconvexfunctiondefinedonaneighborhoodofapointx0insomenormedlinearspaceandcontinuousatx0.Showthatthereexistanx()・neighborhoodVandapositiveconstant2suchthatwheneverxandybelon

8、gtoVwehavetheLipschitzinequality

9、/(x)—f(y)W2

10、

11、x—y\.(Itmaybeassumedthatx0=0;choose5>0sothat

12、/(x)—/(0)

13、<1if

14、

15、x

16、

17、<§・Thenwemaytake7=~U(X)and厶2=8/5.)ItfollowsthattherestrictionofacontinuousconvexfunctiontoacompactconvexsetinXsatisfiesaLipschitzconditionuni

18、formlyonthatset.1.设函数/在DuR”上可微，则于在£>上为凸函数的充要条件是，对任意的x,yg£>,/(y)>/U)4-V/(x)r(y-x).9.Areal-valuedfunctionfdefinedonalinearspaceXisquasi-convexifitssublevelsets{xeX:f(x)W2}areconvexforeachreal2.Showthatfisquasi-convexifandonlyiff(tx