My Math Forum A problem in functional Analysis (Difficult)

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 October 10th, 2008, 02:59 AM #1 Newbie   Joined: Oct 2008 Posts: 12 Thanks: 0 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 $J_a$ is a maximal ideal. 3) Inversly, if J is amaximal ideal of C(X), then there existsa in X s.t J=$J_a$. 4) Prove that $\overline{J}$={f C(X)| f(x)=0 , for all x in Z}
 October 11th, 2008, 06:51 PM #2 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 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 --------------------------------------------------- ...
 October 14th, 2008, 07:38 PM #3 Newbie   Joined: Oct 2008 Posts: 12 Thanks: 0 Reponse The most difficult problem is problem 4). All others are trivial.
 October 17th, 2008, 12:04 AM #4 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 Re: A problem in functional Analysis (Difficult) Maybe,it is related to Banach Algebra.
 October 19th, 2008, 08:43 AM #5 Newbie   Joined: Oct 2008 Posts: 12 Thanks: 0 Re: A problem in functional Analysis (Difficult) Just try with the quotent topology

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