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Math Help - compactness of a set

  1. #1
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    compactness of a set

    Let F be a subset of C1([0,1]) with metric coming from the C1 norm fC1=f∥∞+ f∥∞. Show that F is compact in C1 if and only if F is closed and bounded and {f: fF} C([0,1]) is equicontinuous.


    could anyone help me to solve this question ?
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    Let F be a subset of C1([0,1]) with metric coming from the C1 norm fC1=f∥∞+ f∥∞. Show that F is compact in C1 if and only if F is closed and bounded and {f: fF} C([0,1]) is equicontinuous.


    could anyone help me to solve this question ?
    This is the Azela-Ascoli Theorem

    A few proofs can be found here. Yay for \frac{\epsilon }{3}

    Arzel?Ascoli theorem - Wikipedia, the free encyclopedia
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  3. #3
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    hi thanks for your help,

    but it seems a bit different from the arzela-ascoli theorem

    in the form of the norm and the original set.

    how do i have to approach when i deal with C1([0,1]) ?
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  4. #4
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    The Ascoli-Arzela can also be stated as:

    A set K \subset C([a,b]) is compact if and only if K is closed, bounded and equicontinuous. In this case, any metric would work as well as long as it's defined on that particular metric space.
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