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Homework Help: Compactness in C([a,b])

  1. Nov 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Let g belong to C([a,b]) be given, and let X={f belonging to C([a,b]) such that abs(f(t))<=abs(g(t)), for any t belonging to [a,b]}

    Find all g's for which X is compact.


    2. Relevant equations
    A compact set is closed and bounded. I believe that all the f's smaller tan g determine a bounded subset, as long as g is bounded. However, i dont understand so muc how can i put the closed part in. I believe that I need to work with the closure, as X will bo closed if X=cl(X), and cl(X)=all the f's smaller than g. However, im not able to put all this together, and to actually answer te question.


    3. The attempt at a solution
    Any suggestion/hint, would be highly thanked.
     
  2. jcsd
  3. Nov 17, 2008 #2

    morphism

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    Compact doesn't mean closed and bounded (boundedness in particular is what isn't enough). We're in C[a,b] here - not some Euclidean space! So I'll let you rethink your approach. If you still need help, post back.
     
  4. Nov 18, 2008 #3
    you're right. quite an important concecptual mistake. i revised my notes, and well, realise that maybe with ascoli-arzela, but im lost. it seems like such a broad question...
    i would need some help if you dont mind. thank you
     
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