How Do You Apply Hahn-Banach Theorem to Extend Functions and Preserve Norms?

[azdang]

Homework Statement

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.

 

[kurt.physics]

Homework Equations f(x)=\alpha_1 \xi_1 + \alpha_2 \xi_2, x=(\xi_1,\xi_2)The Attempt at a Solution I'm not sure where to begin with this problem. I know that I need to find the linear extensions of f and then calculate the norms of both f and the extension. But I have no idea how to start.