About uniqueness in Hahn-Banach Theorem (Foguel's implication proof) I'm reading this note of Foguel: On a Theorem By A. E. Taylor. There, he proves the following

Theorem: Let B be a complex normed space whose conjugate space is not
strictly convex. Then there exists a bounded linear functional on a
subspace of B for which a norm-preserving linear extension to B is not
unique.

I have three questions regarding the proof:

*

*Why does $||(f_1 +f_2)/2||=1$ is true? I know that there must exists some $\alpha \in (0,1)$ such that $||\alpha f_1 +(1-\alpha)f_2||=1$; however, why can we choose $\alpha=1/2$?

*Why does $\lim\limits_{n \to \infty} (f_1(x)+f_2(x))/2=1$ implies $\lim\limits_{n \to \infty} f_1(x)=\lim\limits_{n \to \infty} f_2(x)=1$?

*Why does $\lim\limits_{n \to \infty} f_2(y_n)=1$?
Thanks in advance. I really appreciate any help.
 A: *

*By failure of strict convexity, there are $g_1 \neq g_2$ and $0 < \alpha < 1$ so that $||g_1|| = ||g_2|| = ||\alpha g_1 + (1-\alpha)g_2|| = 1.$
We claim that for all $\beta \in [0, 1],$ we have $||\beta g_1 + (1-\beta)g_2|| = 1,$ which in particular means you can take $\beta = 1/2.$
The idea is simple: Assume not. Let $\beta \in [0, 1]$ be so that $f = \beta g_1 + (1-\beta)g_2$ has $||f|| < 1.$
Since $f$ is on the segment between $g_1, g_2,$ we know that $\alpha g_1 + (1-\alpha)g_2$ is either a convex combination of $f$ and $g_1$ or a convex combination of $f$ and $g_2.$ Either way, triangle inequality deduces a problem since $||f|| < 1.$ (For these types of problems, just draw a picture of the $\ell^1$ or $\ell^{\infty}$ norms on $\mathbb{R}^n$ and you'll see what to do.)


*Suppose $f_1(x_n)$ didn't tend to $1.$ Then it has to have some subsequence converging to something smaller than $1,$ by the fact that $||f_1|| = 1$ and Bolzano-Weierstrass. It's easy to see that along this subsequence, $f_2(x_n)$ must converge to something bigger than $1.$


*As $n\rightarrow\infty,$ $|x_n-y_n|$ tends to $0$ and bounded linear functions are uniformly continuous.
