A question in the proof of $C(X)$ is not a dual space of a Banach space.

The set of constant functions $\{f\equiv a: a\in[-1,1]\}$ is already weak*-closed (for instance, it is compact, since the map $[-1,1]\to C_\mathbb{R}(X))$ sending $a$ to the constant function taking value $a$ is continuous and $[-1,1]$ is compact), so you've shown that in fact every element of $Ball(C_\mathbb{R}(X))$ is a constant function. This means every continuous function $X\to\mathbb{R}$ is constant. But by Urysohn's lemma, if $x,y\in X$ are distinct points, there is a continuous function $f:X\to\mathbb{R}$ such that $f(x)=0$ and $f(y)=1$. Such a function is not constant, so it must be impossible to have distinct points $x,y\in X$. That is, $X$ must have only one point.