= Solution
\b[True.] Choose a countable dense sequence $(x_j)$ in the separable <Banach space> $E$. For $\phi,\psi$ in its closed dual <unit ball>, put
$$
\rho(\phi,\psi)=\sum_{j=1}^\infty2^{-j}
\frac{|(\phi-\psi)(x_j)|}{1+|(\phi-\psi)(x_j)|}.
$$
This is a <metric>: positivity follows because continuous functionals agreeing on a <dense subset> agree everywhere; symmetry is immediate; and the triangle inequality follows from subadditivity of $t/(1+t)$ for $t\geq0$. The tail of the series is uniformly small, so convergence on these coordinates implies convergence for this metric, and conversely metric convergence controls each individual coordinate.
The uniform bound $\|\phi-\psi\|\leq2$ upgrades coordinate convergence to <pointwise convergence> on all of $E$. For any $x\in E$, choose $x_j$ close to $x$ and use
$$
|\phi(x)-\psi(x)|\leq|\phi(x_j)-\psi(x_j)|+2\|x-x_j\|.
$$
For neighbourhoods involving finitely many $x$, choose finitely many such approximations. This proves equality of the induced topologies, including for arbitrary nets. Thus the <weak-star metrizability of the dual ball> gives
$$
\boxed{B'\text{ is metrizable in its relative weak-star topology}.}
$$
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