= Solution
The dual $X^*$ is the <vector space> of bounded <linear maps> $f:X\to\mathbb F$, with norm
$$
\lVert f\rVert=\sup_{\lVert x\rVert\leq1}|f(x)|.
$$
If $(f_n)$ is Cauchy in this norm, then $(f_n(x))$ is Cauchy in the complete <scalar> field for every $x$. Define $f(x)=\lim_nf_n(x)$. Pointwise passage to the <limit> preserves <linearity>. Given $\varepsilon>0$, choose $N$ such that $\lVert f_n-f_m\rVert<\varepsilon$ for $m,n\geq N$; sending $m\to\infty$ gives
$$
|f_n(x)-f(x)|\leq\varepsilon\lVert x\rVert.
$$
\b[Thus $f$ is bounded and $\lVert f_n-f\rVert\leq\varepsilon$. Hence $X^*$ is Banach, even if $X$ was only normed.]
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