Thank you so much. However, I am also looking for an $l_2$ space with an equivalent norm such that the norm is smooth but not strictly convex.

In the same question, if we take $\Vert x_n + x \Vert \to 2$, for $x, x_n \in S_X$, it will imply that $\exists f \in J(x)$ such that $f(x_n) \to 1$. But does the above property is true for each $f \in J(x)$?

Yes sir, sorry about the confusion.

I am sorry to interrupt @Tanmoy Paul. In Example 3.12. [Jayanarayanan, C. R., and Tanmoy Paul. "Strong proximinality and intersection properties of balls in Banach spaces." Journal of Mathematical Analysis and Applications 426.2 (2015): 1217-1231.], it has been found that $Y=kerf$ is not strongly proximinal in $l_1$. Could you please clarify your answer? Thank You.

Ok, sir, thank you.

Could you check whether my attempt is right or wrong? I claim that Y is not ball proximinal, for which I have used theorem 3.2. from (Ball proximinal spaces, Indumathi, Lalithambigai, J. Convex Analysis, {vol 18}(2), 353--366, (2011)). By taking x=(0, 2, 0, ...) $\in l^1$, it satisfies the theorem conditions, but $P_{J_X(f)}(x)=\phi $ can be found. So Y is not ball proximinal.

Thank you so much, sir, for your valuable comments. I will work on it.

I am trying to find an element in $B_Y$ according to the definition, but I can't find any element in this set. And so I need help with this problem @Sam Sanders.