Equicontinuity of compact families of characters (${\sup }_{\chi \in C}|\chi (x)-1|⟶0(x⟶e)$)

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 $\overline{\chi (x)}=\widehat{T_{x}a}(\chi )/\hat{a}(\chi )$ 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.