1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted