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Math Help - Norms on Functionals 1

  1. #1
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    Norms on Functionals 1

    Dear Colleagues,

    Could you please help me in solving this problem:
    Find the norm of the linear functional f defined on C[-1,1] by f(x)=\int_{-1} ^{0} x(t)dt-\int_{0} ^{1} x(t)dt.
    I have already proved that ||f||\leq 2, it remain to show that ||f||\geq 2.
    Remark ||x||=max \ x(t), t\in [-1,1].

    Regards,

    Raed.
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  2. #2
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    Quote Originally Posted by raed View Post
    Dear Colleagues,

    Could you please help me in solving this problem:
    Find the norm of the linear functional f defined on C[-1,1] by f(x)=\int_{-1} ^{0} x(t)dt-\int_{0} ^{1} x(t)dt.
    I have already proved that ||f||\leq 2, it remain to show that ||f||\geq 2.
    Remark ||x||=max \ x(t), t\in [-1,1].

    Regards,

    Raed.
    Sorry, ||x||=max \ |x(t)|, t\in [-1,1].
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  3. #3
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    Try taking x(t) = -t^{1/n}, where n is an odd integer (so that x(t) is defined when t is negative).

    The idea is that you want x(t) to be close to –1 when t is positive, and close to +1 when t is negative. But x has to be a continuous function, so it will have to change rapidly as t goes from negative to positive. Another choice for x(t) would be to define it to be +1 in the interval [–1,–1/n], –1 in the interval [1/n,1], and x(t) = –nt in the interval [–1/n,1/n].
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  4. #4
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    Thank you very much for your reply.

    Regards,

    Raed.
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