= Weak-star topology on an entire infinite-dimensional Banach dual is not metrizable
A countable local base at zero would give countably many finite sets of evaluation vectors. Any other evaluation must lie in the span of one of those finite sets: otherwise a bounded functional vanishing on that set but not on the evaluation vector, supplied by <Hahn-Banach theorem>, contradicts inclusion of neighbourhoods after scalar rescaling. Thus the predual has countable Hamel dimension, impossible for an infinite-dimensional <Banach space> by the <Baire category theorem>. This concerns the entire dual, unlike <weak-star metrizability of the dual ball>.
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