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Math Help - Unbounded functional

  1. #1
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    Unbounded functional

    Dear Colleagues,

    Could you please help me to solve the following problem:

    The space C^{1}[a,b] is the subspace of C[a,b] consists of all continuously differentiable functions. Let f be a functional defined on C^{1}[a,b] given by f(x)=x^{'}(c),c=(a+b)/2 where x\in C^{1}[a,b]. Prove that f is not bounded.

    Regards,

    Raed.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help me to solve the following problem:

    The space C^{1}[a,b] is the subspace of C[a,b] consists of all continuously differentiable functions. Let f be a functional defined on C^{1}[a,b] given by f(x)=x^{'}(c),c=(a+b)/2 where x\in C^{1}[a,b]. Prove that f is not bounded.

    Regards,

    Raed.
    I think you can do this on your own. Think about it, what if you created your function to be such that for every \varepsilon>0 you create a function f_\varepsilon\in C^1[a,b] such that \displaystyle f'_\varepsilon\left(\frac{a+b}{2}\right)=\frac{1}{  \varepsilon}
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  3. #3
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    Thank you very much.
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