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