27.1 Locally Compact Groups
Definition 27.1.1 (Locally Compact Group).label Let $G$ be a topological group, then $G$ is locally compact if $G$ is a LCH space.
Proposition 27.1.2.label Let $G$ be a locally compact group, $E$ be a TVS over $K \in \RC$, and $\phi \in C_{c}(G; E)$, then $\phi$ is left and right uniformly continuous.
Proof. By Lemma 5.20.2, there exists a compact neighbourhood $U \in \cn_{G}(\text{supp}(\phi))$. By Proposition 6.4.5, $\phi|_{U}$ and $\phi|_{\supp(\phi)^c}$ are both left and right uniformly continuous. Therefore $\phi$ is left and right uniformly continuous.$\square$
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