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Math Help - Compactness

  1. #1
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    Compactness

    Hello,
    I need help with the following questions:
    Let B = {f in C([0,1], R) | f' is continuous on (0,1), f(0)=0, and |f'(x)| <= 1}. Show that cl(B) is compact.
    I'm stuck on this problem. Appreciate anyone's help on this. Thank you in advance. Thanks
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  2. #2
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    Quote Originally Posted by anlys View Post
    Hello,
    I need help with the following questions:
    Let B = {f in C([0,1], R) | f' is continuous on (0,1), f(0)=0, and |f'(x)| <= 1}. Show that cl(B) is compact.
    I'm stuck on this problem. Appreciate anyone's help on this. Thank you in advance. Thanks
    Just remember that for a C^1 function they are equivalent:

    1) f is Lipschitz cont. with constant M
    2) |f'| is bounded by M

    Use this to prove that \delta = \frac{ \epsilon }{M} works in the def. of (uniform) equicontinuity of B. Now use Arzela-Ascoli.
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  3. #3
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    Mean value theorem shows that B is equicontinuous .

    Moreover, since f(0)=0, again mean value shows that, for x\in (0,1)
    |f(x)|=|f(x)-f(0)|\leq |x-0|=|x| for each f\in B. This yields that B is pointwise bounded (for x=1 is an easy consequence). Now Arzela Ascoli tells us that B is relatively compact, that is equivalent to the desired asertion.

    By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post
    Last edited by Enrique2; January 18th 2010 at 01:39 PM. Reason: didn't see Jose27 post
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  4. #4
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    Quote Originally Posted by Enrique2 View Post
    Mean value theorem shows that B is equicontinuous .

    Moreover, since f(0)=0, again mean value shows that, for x\in (0,1)
    |f(x)|=|f(x)-f(0)|\leq |x-0|=|x| for each f\in B. This yields that B is pointwise bounded (for x=1 is an easy consequence). Now Arzela Ascoli tells us that B is relatively compact, that is equivalent to the desired asertion.

    By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post
    Thank you, Jose27 and Enrique2 for the input. I have tried using the hints that you gave me and now I can prove that the set B is equicontinuous and pointwise bounded. Thanks again
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  5. #5
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    Quote Originally Posted by anlys View Post
    Thank you, Jose27 and Enrique2 for the input. I have tried using the hints that you gave me and now I can prove that the set B is equicontinuous and pointwise bounded. Thanks again
    Please excuse me, it is my fault, but you have to check uniform boundedness instad of pintwiese boundedness. Anyway the given argument shows that B is in the unit ball of C([0,1])
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