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$f(x,y)=3x$ on $X=${$(x,y): x=y$} is a linear functional on the subspace $X\subset \Bbb R^2$.

Here the norm on $\Bbb R^2 $ is defined as $\|(x,y)\|= |x|+|y|$. Let $g(x,y)=ax+by$ on $\Bbb R^2$, where $g$ is a Hahn Banach extension of $f$. Then what must $a-b$ be?

I know that if $g$ is restricted to $X$, then $g$ must be equal to $f$ and this implies $a+b=3$. But how do I proceed further?

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    What is the norm of $f$? What are the norms of $g\lvert_{\mathbb{R}\times \{0\}}$ and $g\lvert_{\{0\} \times \mathbb{R}}$? – 2017-02-05
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    norm of should be 3 – 2017-02-05
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    @Upstart: The norm of $f$ is ${3 \over 2}$. What is $\|(x,x)\|$? – 2017-02-05
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    i have actually just started this topic so i am a little unsure.$2|x|$ – 2017-02-05
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    You have $|f(x,x) | = 3|x| = {3 \over 2} 2 |x| = {3 \over 2} \|(x,x)\|$, from which you can read off the operator norm of $f$. – 2017-02-05
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    You can compute the norm of $g$ in terms of $a,b$. Then you have the additional constraint in $a,b$ that you discovered. Then an 'ah hah' moment :-). – 2017-02-05

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