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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 $\frac{1}{\sqrt 5}$.

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

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