I had a doubt on Hahn-Banach continuous extension theorem. Wikipedia says this:
Hahn–Banach continuous extension theorem — Every continuous linear functional $f$ defined on a vector subspace $M$ of a (real or complex) locally convex topological vector space $X$ has a continuous linear extension $F$ to all of $X.$ If in addition $X$ is a normed space, then this extension can be chosen so that its dual norm is equal to that of $f.$
Say I am working in $(\mathbb{R}^2,||\cdot ||_1).$ Here, for $(x,y) \in \mathbb{R}^2,$ we have $||(x,y)||_1=|x|+|y|.$ So, any linear functional on the space $( \mathbb{R}^2,|| \cdot ||_1 )$ will reside in $(\mathbb{R}^2,|| \cdot ||_{\infty}),$ as dual norm of $1-$ norm is $\infty-$ norm.
But, while reading this answer in MSE, I am confused as to how the norm for the extension is taken. First point to note is that $g$ lives in $\infty-$ norm. As for the functional $g,$ we need $||g||=3= ||(a,b)||_{\infty}=\max \{|a|,|b| \}.$
So, either of $|a|$ or $|b|$ must be $3.$ Both can be $3$ also. But the answer in the linked part says that $a \leq \frac{3}{2}$ and $b \leq \frac{3}{2}$ making $||g|| \leq \frac{3}{2},$ a contradiction.
How is this achieved here ? Whats happening here ??
Also, the solution given to the question involves simple geometry which I am not able to follow. Is there any standard procedure/algorithm to find the Hahn-Banach extension given a space, its subspace and a functional on the subspace with all the necessary assumptions satisfied ? please let me know of the simple references also.
New edit: Using the notation in the linked question, I wrote (above in this question itself) that $||g||_\infty=3$ thinking that $||f||_{\infty}=3$ in $\mathcal{B}(M,\mathbb{R}).$
I have made a mistake there. $||f||_{\infty}=3$ is wrong. In fact $||f||_{\infty}=\frac{3}{2}.$ Indeed, $$||f||_{\infty} \text{ (in $M$ )}=\sup_{|(x,x)| \ne 0} \frac{|f(x,x)|}{||(x,x)||_1}= \sup_{|(x,x)| \ne 0} \frac{|3x|}{2|x|}=\frac{3}{2}.$$
So, we have that $||g||_\infty = \frac{3}{2}.$
And the rest of the work given in the linked answer is alright to me. Can someone verify this for me, please ? Whether there is any gap in my understanding ?
Please help.
Thanks in advance.