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Math Help - Question regarding Equicontinuity and the open ball of functions

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    Question regarding Equicontinuity and the open ball of functions

    Hey - my hunch tells me that given an arbitrary open ball of functions (such as an arbitrary open ball centered at some f*) in the space of continuous functions defined on the compact interval [a,b] (ie, C0[a,b] sorry I don't know latex very well) with the standard sup norm, that this set is NOT an equicontinuous set of functions. Is this hunch true?? Anyone know how to prove this???
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frederick111 View Post
    Hey - my hunch tells me that given an arbitrary open ball of functions (such as an arbitrary open ball centered at some f*) in the space of continuous functions defined on the compact interval [a,b] (ie, C0[a,b] sorry I don't know latex very well) with the standard sup norm, that this set is NOT an equicontinuous set of functions. Is this hunch true?? Anyone know how to prove this???
    My first inkling would to use Ascoli-Arzela to prove that if B_{\varepsilon}(f) were equicontinuous (and hopefully) bounded then it would be compact. But, every compact subspace of a Hausdorff space is closed contradicting that it's open.

    This is just a very first impression.
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    Thanks Drexel - that wouldn't work though because Be(f) is an open ball and in order to claim compactness using Arzela-Ascoli we would need the additional property that the set is closed, which of course is not the case for the open ball....
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by frederick111 View Post
    Thanks Drexel - that wouldn't work though because Be(f) is an open ball and in order to claim compactness using Arzela-Ascoli we would need the additional property that the set is closed, which of course is not the case for the open ball....
    Ah! Dang, hypotheses!

    I'll give this a thought...in fact...I think I am already supposed to be thinking about this ...see. I'll actually give thought to it now!
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    Let B be the open ball of radius r around f

    Equicontinuity implies for all g in the open ball, and for all x in [a,b] there exists an d s.t.

    lx-yl < d --> lg(x)-g(y)l < r/2

    (here I set e = r/4)

    but, I think it very possible to make a function h where h is in the open ball of radius r around f, but does not satisfy the above condition.

    All you would have to do is ensure that on some interval of size d, you have h(x) = f(x)+r/2 and h(y) = f(y)-r/2, where f(x) >= f(y), cause then

    lg(x)-g(y)l = lh(x)-h(y)+r l = h(x)-h(y)+r (since both terms are +)

    h(x)-h(y)+r>= r > r/2
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