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October 10th, 2008, 02:59 AM   #1
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A problem in functional Analysis (Difficult)

We denote by C(X,R) is the set of continuous functions from X to real number set R
Let X be a compact metric space and J be an ideal of the ring (C(X,R),+,.)
Denote by Z={x X| g(x)=0 for all g in J}
1) Prove that if Z is an empty set then J contains a function g>0, i.e g(x)>0 for all x in X. Therefore, J=C(X).
2) For a X, we set .Prove that is a maximal ideal.
3) Inversly, if J is amaximal ideal of C(X), then there existsa in X s.t J=.
4) Prove that ={f C(X)| f(x)=0 , for all x in Z}
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October 11th, 2008, 06:51 PM   #2
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Re: A problem in functional Analysis (Difficult)

Z is an empty set
x:X==>exist(g:J).g(x)>0
ex(g:J) . g>0 on N(x) ;; here,N(a) is a neighbor-set of x
there must be a finit set-set {Ni|i=1:n} gi>0 on Ni
note g=sum(gi) is the very function
----------------------------------------------------
find a f(a)!=0 and g:Ja make sure f+g>0
---------------------------------------------------
...
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October 14th, 2008, 07:38 PM   #3
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The most difficult problem is problem 4).
All others are trivial.
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October 17th, 2008, 12:04 AM   #4
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Re: A problem in functional Analysis (Difficult)

Maybe,it is related to Banach Algebra.
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October 19th, 2008, 08:43 AM   #5
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Re: A problem in functional Analysis (Difficult)

Just try with the quotent topology
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