# Hahn-Banach Theorem (Normed Spaces)

• May 2nd 2011, 04:45 AM
raed
Hahn-Banach Theorem (Normed Spaces)
Dear Colleagues,

Best Regards.

Raed.
• May 2nd 2011, 08:20 AM
Opalg
Since $\displaystyle \mathbb{R}^n$ is an inner-product space, every linear functional is given by an inner product. In fact, if $\displaystyle \alpha = (\alpha_1,\alpha_2)$ then the functional $\displaystyle f(x) = \alpha_1\xi_1+\alpha_2\xi_2$ is given by $\displaystyle f(x) = \langle x,\alpha\rangle$. Use that fact to show that $\displaystyle \|f\|_{\mathbb{R}^2} = \sqrt{\alpha_1^2+\alpha_2^2}$.

Now that you know $\displaystyle \|f\|$, you have to figure out how to extend f to a linear functional on $\displaystyle \mathbb{R}^3$ without increasing the norm.
• May 2nd 2011, 10:36 AM
raed
Thank you very much for your reply I have already proved the norm of f. But the problem is in finding a linear extension of f that preserves the norm.

Best Regards.
• May 2nd 2011, 12:04 PM
Opalg
Quote:

Originally Posted by raed
I have already proved the norm of f. But the problem is in finding a linear extension of f that preserves the norm.

So the extension to $\displaystyle \mathbb{R}^3$ must be of the form $\displaystyle \tilde{f}(x) = \alpha_1\xi_1+\alpha_2\xi_2+\alpha_3\xi_3$, and its norm will be $\displaystyle \|\tilde{f}\| = \sqrt{\alpha_1^2+\alpha_2^2+\alpha_3^2}.$ That leaves only one choice for $\displaystyle \alpha_3.$