I consider $\ell^2$ over real numbers. Define the equivalent norm $\|\cdot\|$ on $\ell^2$ by $$\|x\|=\max\{\|x^{\prime}\|,\|x^{\prime\prime}\|\},$$ where for all $x=(x_n)\in \ell^2$, $x^{\prime}=(0,x_2,x_3,\ldots)$ and $x^{\prime\prime}=(x_1,0,x_3,\ldots)$.
I know that $(\ell^2,\|\cdot\|)$ is a reflexive space, having Radon-Riesz property but is not strictly convex. Thus $(\ell^2,\|\cdot\|)^*$ can not be smooth. I want to know is $(\ell^2,\|\cdot\|)$ a smooth space? I could not show it neither I could show that it not so. Regarding smoothness, I know that $X$ is smooth iff $\|.\|$ is Gateaux differentiable at all $x\in X$. But could not solve my problem.
There is another issue. For any $x\in S_X$ and for any $(x_n)\subset S_X$ with $\|x_n+x\|\to 2$, I want to check if for $f\in S_{X^*}$, there exists a subsequence $(x_{n_k})$ of $(x_n)$ such that $f(x_{n_k})\to 1$. The first problem I encounter is that I do not know the dual norm. Any help in this regard is appreciated.
Your space is not smooth. In fact, it is easy to check that the norm is not Gateaux-differentiable at the point $x=(1,-1,0,0,\dots )$, because if we take $h=e_1$, then with $x+th$ we have $x'=(0,-1,0,0,\dots)$ and $x''=(1+t,0,0,\dots)$, so $\|x+th\|=\max\{|1+t|,1\}$. Hence for sufficiently small $t$, $$ \frac{\|x+th\|-\|x\|}{t} = \begin{cases} \frac{1+t-1}{t}=1, & \text{if $t>0$} \\ \frac{1-1}{0}=0, & \text{if $t<0$} \end{cases} $$ so that the limit as $t\to 0$ does not exist.