= Solution
<Goldstine theorem> states that the canonical image $J(B_X)$ of the closed unit ball of a <normed vector space> $X$ is weak-star dense in $B_{X^{**}}$.
The <Banach-Alaoglu theorem> states that $B_{X^*}$ is compact in the <weak-star topology>. To prove it, map each $f\in B_{X^*}$ to its values in
$$
P=\prod_{x\in X}\{z\in\mathbb C:|z|\leq\lVert x\rVert\}.
$$
Every factor is compact, so <Tychonoff theorem> makes $P$ compact. The image of $B_{X^*}$ is cut out by the closed linearity conditions
$$
f(x+y)=f(x)+f(y),qquad f(\alpha x)=\alpha f(x),
$$
and is therefore closed in $P$. The product topology restricted to this image is exactly pointwise convergence on $X$, namely the weak-star topology. Hence the ball is compact.
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