This statement is true. If $(x_n)$ is dense in $B_X$, then on $B_{X^*}$ the weak-star topology is induced by a metric such as
$$
d(f,g)=\sum_{n\geq1}2^{-n}
\frac{|(f-g)(x_n)|}{1+|(f-g)(x_n)|}.
$$
The <Banach-Alaoglu theorem> makes this metric space compact, and every compact metric space is sequentially compact.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 106 / 2 / a / 1 / Solution

 Solution

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