= Solution
\b[True.] In the <norm topology> on the unit ball, the set $\{x\in B:\|x\|<1/2\}$ is a neighbourhood of zero. If the relative <weak topology> agrees with it, there are finitely many <continuous linear functionals> $f_1,\ldots,f_m$ and positive $\epsilon_j$ such that
$$
\{x\in B:|f_j(x)|<\epsilon_j\text{ for }1\leq j\leq m\}
\subseteq\{x\in B:\|x\|<1/2\}.
$$
If $\bigcap_j\ker f_j$ contained a nonzero vector, rescaling it to have norm $3/4$ would put it in the left-hand set but not the right-hand set. Therefore $\bigcap_j\ker f_j=\{0\}$. The <linear map>
$$
x\longmapsto(f_1(x),\ldots,f_m(x))
$$
is injective into a finite-dimensional scalar space, so
$$
\boxed{\dim E\leq m<\infty.}
$$
This proof works for either the open or closed unit-ball convention and does not require completeness.
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