Let X be a Banach space. If $D \subset X^*$ is (weak*ly or strongly?) dense, then does $f(x_n) \to f(x)$ $\forall f \in D$ imply that $x_n \to x$ weakly?
My thoughts: If $g_m \to g$ in the dual, then we can write:
$|g(x) - g(x_n)| \leq |g(x) - g_m(x)| + |g_m(x) - g_m(x_n)| + |g_m(x_n) - g(x_n)|$, where the middle term always vanishes for a fixed $m$ as $n \to \infty$ by assumption, though this is a moving target. The first term will vanish as $m \to \infty$ in either the weak* topology or the strong topology on the dual, and the latter term vanishes if the dual topology was the strong topology and the sequence $x_n$ was bounded in norm.