= Solution
The <dual space> $X'$ consists of bounded linear maps $g:X\to\mathbb C$, with <operator norm> $\|g\|=\sup_{\|x\|\leq1}|g(x)|$. It is Banach even if $X$ is not: an operator-<norm> <Cauchy sequence> has pointwise limits, these define a <bounded linear functional>, and the uniform Cauchy estimate on the unit ball gives operator-<norm> convergence.
Restriction $R(g)=g|_Z$ is linear and contractive. If $Z$ is <dense>, for every $x\in X$ choose $z_n\in Z$ with $z_n\to x$. The bound $|g(z_n)|\leq\|Rg\|\|z_n\|$ passes to the limit, proving $|g(x)|\leq\|Rg\|\|x\|$. Hence $\|g\|=\|Rg\|$ and restriction is <injective>. Conversely, for $h\in Z'$ define $g(x)=\lim h(z_n)$. The estimate $|h(z_n)-h(z_k)|\leq\|h\|\|z_n-z_k\|$ gives existence and independence of the approximating sequence. It also gives linearity and $\|g\|\leq\|h\|$. Thus \b[restriction is an onto linear <isometry> of dual spaces].
For the sequence subspace in this part, the <norm> is the unweighted $\ell^1$ <norm>; only its membership condition uses the weight $j$. It contains all finitely supported sequences, hence is <dense> in $\ell^1$. Applying restriction to $X=\ell^1$ and using its known dual yields
$$
\boxed{Z'\cong\ell^\infty\text{ isometrically},\qquad h_b(\alpha)=\sum_{j=1}^\infty b_j\alpha_j,\quad\|h_b\|=\|b\|_\infty.}
$$
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