Thank you Jochen for pointing out. I was a bit confused about Hahn-Banach smoothness and smoothness. A Banach space $X$ is said to be Hahn-Banach smooth if every $x^*\in X^*$ has unique norm preserving extension to $X^{**}$. Similarly a weaker version of this property can also be defined for the norm attaining functionals. If $X^*$ is strictly convex then any subspace of $X$ is Hahn-Banach smooth BUT this property does not imply (weakly) Hahn-Banach smoothness of $X$.

Yes, but I need an example with Real scalar. The space $C(X)$ I considered with Real scalars.

I would like to add one comment and one question in the same thread. RNP is closely related to Asplund property. A Banach space $X$ is said to be an Asplund space if every continuous convex function is Frechet differentiable in a dense $G_\delta$ set. Again it has many equivalent characterizations like $X$ is Asplund if and only if every equivalent norm on it has at least one point of Frechet differentiability also, $X$ is Asplund iff $X^*$ have RNP. It is known that if $X$ has an equivalent norm which is Frechet diff then $X$ is Asplund. My question is whether the converse is also true?

Yes, I mean, whether $\|T_n(f)-f\|_\infty\to 0$ for all $f\in A$ implies $\|T_n(g)-g\|_\infty\to 0$?