I am trying to understand the Remark 3.2 mentioned in the paper titled as "On Weak* -Extreme Points in Banach Spaces" written by S. Dutta and T. S. S. R. K. Rao (http://library.isical.ac.in:8080/jspui/bitstream/10263/3039/1/on%20weak%20extreme.pdf).
I am not able to understand the statement: If $\Lambda \in \partial_eX_1^{***}$ is a point of weak* - weak continuity for the identity map on $X_1^{***}$ then again by the denseness of $X_1^*$ in $X_1^{***}$ we have that $\Lambda = x^*\in \partial_eX_1^{*}$.
Here, $X_1$ refers to the closed unit ball of the Banach space $X$ and $\partial_eX_1$ refers to the set of extreme points of $X_1$. $X_1^*$ refers to the closed unit ball of the dual space of $X$.
Also, I am having difficulty understanding the following example from the same paper.
$\textbf{Example:}$ Let $K$ be a compact set and $k_0\in K$ be an accumulation point. Since $\chi_{k_0}\in C(K)^{**}$ it is easy to see that $\delta (k_0)\in \partial_e C(K)_1^*$ is not a point of weak-weak continuity for the identity map on $C(K)_1^*$. However, since $\delta (k_0)$ is a denting point, it is a weak-denting point of $C(K)_1^{***}$ and hence is a point of weak-weak (in fact weak-norm) continuity for the identity map on $C(K)_1^{***}$.
I am not able to understand how to show a point is a weak*-weak continuous point for the identity map defined on the said domain.
How easily they can conclude a point is not a weak*-weak continuous point in the above example.
How to conclude a weak* denting point becomes a point of weak*-weak continuous point for the identity map in the above example?
Please help me. Any help is appreciated. Thank you in advance.
