# Hahn Banach extensions.

Consider the normed linear space $$R^2$$ equipped with the norm given by $$||(x,y)||=|x|+|y|$$ and the subspace $$X=\{(x,y)\in R^2 : x=y\}.$$ Let $$f(x,y)=3x$$ on $$X$$. Then what is the Hahn Banach extension of $$f$$.?

I have gone through different books and dufferent methods and idea of how to find such extensions. But stil not comfortable with it. Could you please help me understand a general method? Is there any general idea? Is the norm of $$f \ 3?$$. A similar question is there in one book but there norm of $$f$$ is $$\frac{1}{\sqrt 5}$$.

• Why do you think it is unique? – user10354138 Sep 30 '18 at 17:45
• I am getting unique extension as $g(x,y)=\frac{3x}{2}+\frac{3y}{2}$. – Ziya Sep 30 '18 at 17:56
• But I don't know whether it is correct or not. – Ziya Sep 30 '18 at 17:58
• another extension is $g(x,y)=3x$ – Aweygan Sep 30 '18 at 22:55
• How to find such extensions? – Ziya Oct 1 '18 at 0:47