= Solution
Throughout this question the <Banach spaces> and <Hilbert spaces> are complex. Separability is an additional standing hypothesis, not part of the general definition of a <Hilbert space>. Here and below <unit ball> means the closed unit ball. An open unit ball in positive finite dimension is not compact. In finite dimension, equivalence of <norms> identifies the closed unit ball with a closed bounded subset of Euclidean space, so the <Heine-Borel theorem> gives compactness in the <norm topology>.
In an infinite-dimensional <Hilbert space>, the <Gram-Schmidt process> constructs an infinite <orthonormal sequence> $(e_n)$. Put $f_n=e_n/\sqrt2$. Then $\|f_n\|=1/\sqrt2\le1$ and, for $m\ne n$, $\|f_n-f_m\|^2=(1+1)/2=1$. No subsequence is a <Cauchy sequence>, contradicting <sequential compactness>. Thus \b[the closed unit ball is norm compact exactly in finite dimension].
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