Hahn-Banach Theorem (Normed Spaces)

**raed** · May 2nd 2011, 04:45 AM

Dear Colleagues,

I have a problem in the attachment and could you please help me in solving it.

Best Regards.

Raed.

**Opalg** · May 2nd 2011, 08:20 AM

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

Now that you know $\|f\|$ , you have to figure out how to extend f to a linear functional on $\mathbb{R}^3$ without increasing the norm.

**raed** · May 2nd 2011, 10:36 AM

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.

**Opalg** · May 2nd 2011, 12:04 PM

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

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