I am to illustrate a particular theorem by considering a functional f on R^2 defined by f(x)=\alpha_1 \xi_1 + \alpha_2 \xi_2, x=(\xi_1,\xi_2), its linear extensions \bar{f} to R^3 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 R^2 where, ||x||=1. And the norm of the extension is the sup| \bar{f}(x)| over all x's in R^3 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.