\section{Locally Compact Groups}
\label{section:lcg}

\begin{definition}[Locally Compact Group]
\label{definition:lcg}
    Let $G$ be a topological group, then $G$ is \textbf{locally compact} if $G$ is a LCH space. 
\end{definition}

\begin{proposition}
\label{proposition:lcg-cc-uc}
    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. 
\end{proposition}
\begin{proof}
    By \autoref{lemma:lch-compact-neighbour}, there exists a compact neighbourhood $U \in \cn_G(\text{supp}(\phi))$. By \autoref{proposition:uniform-continuous-compact}, $\phi|_{U}$ and $\phi|_{\supp(\phi)^c}$ are both left and right uniformly continuous. Therefore $\phi$ is left and right uniformly continuous. 
\end{proof}
