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Let $E$ be Banach, $\mu_n\to\mu$ weakly on a locally compact $X$, and $f \in C_b(X, E)$. Does $\int f\mathrm d\mu_n\to\int f\mathrm d\mu$ in norm?

Let $X$ be a metric space, $(E, |\cdot|)$ a Banach space $\mathcal M(X)$ the space of all finite signed Borel measures on $X$, $\mathcal C_b(X)$ be the space of real-valued bounded continuous ...
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