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felix2018

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[求助] 各位学泛函的大神们请进！有六道小题希望可以帮忙！已有2人参与

此六道题都是泛函分析的习题，希望各位大神可以帮帮忙！（

由于自己输入的问题，所以在括号做了个解释！

）
1.show that if x is compact,then given ε<0，there exist a finite set of points{xi}∈X,(i为下标)can be approximated to within ε by one of the xi for each x∈X,there is an xi such that |x-xi|<=ε
2.if Ωis bounded,what is the completion of Cc0 (空间下标c上标0)in the supremum norm?deduce that C2 0(空间下标2上标0)is not a banach space with this norm,treat similarly the case Ω=R m(m维欧式空间)
3.show that given s∈S(Ω),1<=p<∞,and ε>0,there exists an f∈C2 0(上标0下标2),such that ||f-s||Lp<ε(下标Lp空间),such that any point x∈X
4.let kㄈH(k属于H),be a convex cone with vertex at 0.i.e,0∈k and λu+μv∈k,any λ,μ>0,any u, v∈k,assume in addition that k is closed.given f∈H,prove that u=Pk f (f在k上的投影)in characterized by the following properties.any u∈k,(f-u,v)<=0,any v∈k and (f-u,u)=0
5.let Ω be a measure space and let h:Ω→[0,∞) be a measurable function .let k={u∈L2(Ω),|u(x)|<=h(x),a.e.on Ω}(2为上标L2空间)check that k is a nonempty closed convex set in H=L2(Ω).determine Pk.
6.let cㄈH be a nonempty closed convex set and let T:c→c,be a monlinear contraction i.e |Tu-Tv|<=|u-v|,any u,v∈c
1)let {un}(n为下标)be a sequence in c such that un→u weakly and |un-Tun|→f Strong prove that u→Tu=f
2)deduce that if c is bounded and T(c)ㄈc,then T has a fixed point.

[ Last edited by felix2018 on 2013-12-21 at 10:19 ]

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