Let us consider the space $\ell_2$ with an equivalent norm defined by $$ \Vert x \Vert = \max \{ \Vert x^{'} \Vert_2, \Vert x^{''} \Vert_2 \}, $$ where $x^{'}=(0, x_2, x_3, \cdots)$, $x^{''} = (x_1, 0, x_3, \cdots)$ and $\Vert \cdot \Vert_2$ is the $\ell_2$-norm. Now, such the norm $\Vert \cdot \Vert$ on $\ell_2$ is not strictly convex. I am trying to prove/discard whether $(\ell_2, \Vert \cdot \Vert)$ satisfies the property:" whenever $x, x_n \in S_X$ satisfying $\Vert x+x_n \Vert \to 2$, then $f(x_n) \to 1$, for all $f \in J(x)=\{ f \in S_{X^*} : f(x)=1\}$." Here $S_X$ is the unit sphere of $(X, \Vert \cdot \Vert)$. I can see that there exists an $f \in J(x)$ satisfying $f(x_n) \to 1$, but whether this holds for all $f \in J(x)$? Thank you.
Here, I have find element $x = (1, 1, 0, \cdots) \in S_X$ and $f = (1, 0, \cdots), g=(0, 1, 0, \cdots)$ $\in$ $J(x)$ that shows that there exists two functionals in $J(x)$. I need to find a sequence $x_n$; if anybody could guide me to find a sequence, I would be obliged. On the positive side, If I could show that $S(X, f, 0)$ and $S(X, g, 0)$ coincide for every $x \in S_X$ and $f, g \in J(x)$; then I will be done. Here $S(X, f, 0)= \{ x \in B_X : f(x) \geq 1\}$ is the slice of the closed unit ball $B_X$.
