= Solution
The domain carries $\sigma(Z^{**},Z^*)$, and the target carries $\sigma(X^*,X)$. For each $x\in X\subseteq Z^*$,
$$
(R\theta)(x)=\theta(x),
$$
which is one of the defining weak-star continuous evaluations on $Z^{**}$. Thus \b[$R$ is weak-star continuous].
Restriction cannot increase a functional's <norm>, so $R(B_{Z^{**}})\subseteq B_{X^*}$. Conversely, any $f\in B_{X^*}$ is a bounded <linear functional> on the <vector subspace>[subspace] $X$ of $Z^*$. The <Hahn-Banach theorem> extends it to a functional $\theta\in(Z^*)^*=Z^{**}$ with the same <norm>. Then $R\theta=f$ and $\|\theta\|\leq1$. Consequently
$$
\boxed{R(B_{Z^{**}})=B_{X^*}.}
$$
This proves exact surjectivity on closed balls, not just density of the image.
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