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Extension of a Functional

  1. Apr 15, 2009 #1
    1. The problem statement, all variables and given/known data
    I am to illustrate a particular theorem by considering a functional f on [tex]R^2[/tex] defined by [tex]f(x)=\alpha_1 \xi_1 + \alpha_2 \xi_2[/tex], [tex]x=(\xi_1,\xi_2)[/tex], its linear extensions [tex]\bar{f}[/tex] to [tex]R^3[/tex] and the corresponding norms.

    I'm having a couple problems with this problem. For one, I haven't ever had to find linear extensions before, so I have no clue how to figure that out.

    The Theorem to apply this to is the Hahn-Banach Theorem for Normed Spaces. I would want to show that the norms of f and the extensions are the same to illustrate this.

    I think the norm of f is the sup|f(x)| over all x's in [tex]R^2[/tex] where, ||x||=1. And the norm of the extension is the sup|[tex]\bar{f}(x)[/tex]| over all x's in [tex]R^3[/tex] where ||x||=1.

    As you can see, I'm pretty lost on most of this. I think I know what I need to figure out, but I just don't have any idea how to get at that. Can anyone offer some guidance? Thank you so much.
     
  2. jcsd
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