
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
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 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} 
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 neighborset of x there must be a finit setset {Nii=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


Tags 
analysis, difficult, functional, problem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Routine proof in Functional Analysis  AfroMike  Real Analysis  6  August 28th, 2013 12:23 PM 
Kakutani Theorem (functional analysis)  PeterPan  Real Analysis  1  April 29th, 2013 11:46 AM 
Functional Analysis: Proposition Proof  Kramer  Real Analysis  2  March 8th, 2013 05:38 AM 
Applications of Functional Analysis  Azari123  Academic Guidance  1  August 11th, 2012 12:29 AM 
A problem from functional analysis  tach  Real Analysis  4  October 20th, 2010 05:53 AM 