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felix2018铁杆木虫 (正式写手)
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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
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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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felix2018
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3楼2013-12-21 12:34:56
liuhaidong
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2楼2013-12-21 12:06:43
感谢参与，应助指数 +1
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4楼2013-12-21 13:02:19
感谢参与，应助指数 +1
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5楼2013-12-21 14:31:15
