Consider a real Banach space $X$ and its continuous dual $X^*$. The Bishop-Phelps Theorem states that the set $$A^*=\{x^*\in X^* \mid x^* \text{ attains its supremum on } \text{ the unit ball } \mathbb{B} \}$$ is norm-dense in $X^*$. Clearly, $\overline{A^*\cap \mathbb{S}^*} \subset \mathbb{S}^*$, with $\mathbb{S}^*$ be the dual unit sphere in $X^*$. Here $\overline{A}$ denotes the norm-closure of $A$. When $X$ is reflexive, $A^*=X^*$ (by James's Theorem) and the equality \begin{equation} \overline{A^*\cap \mathbb{S}^*} = \mathbb{S}^* \label{1}\tag{1} \end{equation} holds.

- Does the equality \eqref{1} holds in arbitrary non-reflexive spaces $X$?
- If the answer is 1. is false, in which non-reflexive space $X$ that the equality \eqref{1} holds?

Thank you for your solutions.