The image of Banach space under its embedding provided by the Banach-Mazur theorem

Question

It is a very nice argument of Banach and Mazur which they use to show that every Banach space $X$ is isometric to a subspace of the space $C(B_{X^*})$, where $B_{X^*}$ is the unit ball of the dual space of $X$. Simply map $x\in X$ to the function $x^* \mapsto \langle x^*, x \rangle$. Can we expect any sort of density (pointwise, weak, etc.) of the image of $X$ in $C(B_{X^*})$?

Answer 1

As Harald Hanche Olsen said, not at all: linear functions are not dense in continuous functions, since linearity is preserved under every conceivable mode of convergence.

For a concrete example, take $X=\mathbb R$. The dual space is also $\mathbb R$, and its unit ball is $[-1,1]$. The space $\mathbb R$ embeds into $C([-1,1])$ via $$x\mapsto \phi_x,\quad \text{ where } \phi_x(t) = xt$$ The image of this embedding is a one-dimensional subspace of $C([-1,1])$.