= Equicontinuity of compact families of characters
{title2=$\sup_{\chi\in C}|\chi(x)-1|\longrightarrow0\quad(x\longrightarrow e)$}
Evaluation $(x,\chi)\mapsto\chi(x)$ is jointly continuous for the <compact-open topology> on the dual of a locally compact group. One proof uses $\overline{\chi(x)}=\widehat{T_xa}(\chi)/\widehat a(\chi)$ near any character where the denominator is nonzero, together with norm-continuity of translations. A finite neighbourhood cover of a compact family $C$ then gives the displayed uniform convergence. This justifies compact-set-and-integrable-tail estimates even when the group is not metrizable and approximation is indexed by a net.
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