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Math Help - Geoemtric interpretation of the norm of a bounded linear functional

  1. #1
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    Geoemtric interpretation of the norm of a bounded linear functional

    Dear Colleagues,

    Could you please help in solving the following problem:
    Show that the norm ||f|| of a bounded linear functional f\neq 0 on a normed space X can be interpreted geometrically as the reciprocal of the distance d=inf\{||x|| \ |f(x)=1\} of the hyperplane
    H_{1}=\{x\in X \ |f(x)=1\} from the origin.

    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 in solving the following problem:
    Show that the norm ||f|| of a bounded linear functional f\neq 0 on a normed space X can be interpreted geometrically as the reciprocal of the distance d=inf\{||x|| \ |f(x)=1\} of the hyperplane
    H_{1}=\{x\in X \ |f(x)=1\} from the origin.

    Regards,

    Raed.
    What have you tried? Where are you stuck? Please make an effort.
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