= Solution
The <weak-star topology> $\sigma(X^*,X)$ is the topology of pointwise convergence on $X$. A basic neighborhood of $f_0$ has the form
$$
\{f\in X^*:|f(x_j)-f_0(x_j)|<\eta,\ 1\leq j\leq m\},\qquad x_j\in X,\quad\eta>0.
$$
Let $J:X\to X^{**}$ be the <canonical embedding into the bidual>, $Jx(f)=f(x)$. It is an isometry. The <Goldstine theorem> says that
$$
\boxed{\overline{J(B_X)}^{\,\sigma(X^{**},X^*)}=B_{X^{**}}.}
$$
Here both balls are closed unit balls. Thus approximation is by finitely many dual evaluations at a time; the theorem does not assert <norm> density in the <bidual space>.
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