Question in proof regarding uniqueness of Hahn Banach extension

Question

proof here

from there http://www.mat.unimi.it/users/zanco/AnFunzionale/pm5_schioppa.pdf

Why is the last equation true? $$\|\frac{f_1+f_2}{2}\|_X=1$$

$f_1$, $f_2$ are extensions of $f$. Does that mean $\frac{f_1+f_2}{2}\Big|_Y=f$? (Is the restriction linear?)

and because $\|f\|_Y=1$ and the norm preserving property of the Hahn Banach extension, we also have $$\|\frac{f_1+f_2}{2}\|_X=1$$

Is this the reason?

Answer 1

$\|\frac {f_1+f_2} 2\|_X \leq \frac {\|f_1\|_X+\|f_2\|_X} 2=\frac { 1+1} 2=1$ and $\|\frac {f_1+f_2} 2\|_X \geq \|\frac {f_1+f_2} 2\|_Y=\|f\|_Y=1$. [The norm on $X$ is greater than or equal to the norm on $Y$: this is immediate from the fact that supremum of a larger set is larger].