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Math Help - Hahn-Banach Theorem (Normed Spaces)

  1. #1
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    Hahn-Banach Theorem (Normed Spaces)

    Dear Colleagues,

    I have a problem in the attachment and could you please help me in solving it.


    Best Regards.

    Raed.
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  2. #2
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    Since \mathbb{R}^n is an inner-product space, every linear functional is given by an inner product. In fact, if \alpha = (\alpha_1,\alpha_2) then the functional f(x) = \alpha_1\xi_1+\alpha_2\xi_2 is given by f(x) = \langle x,\alpha\rangle. Use that fact to show that \|f\|_{\mathbb{R}^2} = \sqrt{\alpha_1^2+\alpha_2^2}.

    Now that you know \|f\|, you have to figure out how to extend f to a linear functional on \mathbb{R}^3 without increasing the norm.
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  3. #3
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    Thank you very much for your reply I have already proved the norm of f. But the problem is in finding a linear extension of f that preserves the norm.


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  4. #4
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    Quote Originally Posted by raed View Post
    I have already proved the norm of f. But the problem is in finding a linear extension of f that preserves the norm.
    So the extension to \mathbb{R}^3 must be of the form \tilde{f}(x) =  \alpha_1\xi_1+\alpha_2\xi_2+\alpha_3\xi_3, and its norm will be \|\tilde{f}\| = \sqrt{\alpha_1^2+\alpha_2^2+\alpha_3^2}. That leaves only one choice for \alpha_3.
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