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

    Now that you know $\displaystyle \|f\|$, you have to figure out how to extend f to a linear functional on $\displaystyle \mathbb{R}^3$ 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 $\displaystyle \mathbb{R}^3$ must be of the form $\displaystyle \tilde{f}(x) = \alpha_1\xi_1+\alpha_2\xi_2+\alpha_3\xi_3$, and its norm will be $\displaystyle \|\tilde{f}\| = \sqrt{\alpha_1^2+\alpha_2^2+\alpha_3^2}.$ That leaves only one choice for $\displaystyle \alpha_3.$
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