Definition 3.7.3 (Translates and Inversions).label Let $G$ be a topological group, $f$ a (complex-valued) function on $G$ and $y\in G$ then left, right translates of $f$ through $h$ and inversion
are homeomorphisms and the maps $y\mto L_{y},R_{y}$ are group homomorphisms. Similarly for each subset $E\suf G$, translates and inversion acts by
We say $f$ is left (resp. right) uniformly continuous if $\norm{L_yf-f}{\sup}\to 0$ (resp. $\norm{R_yf-f}{\sup}$). If $f\in C_{c}(G;\com)$ then $f$ is left and right uniformly continuous.
Proof. Since $G$ is a group, translates and inversion are bijective and by Definition 3.7.1(2) they are homeomorphisms. For homomorphisms
and similarly for right translate.
Let $x\in K$, $\exists U_{x}\in\cali{N}(1)$ such that $\abs{f(xy)-f(x)}<\frac{\e}{2}$ for $y\in U_{x}$ hence there exists symmetric neighborhood $V_{x}$ of 1 such that $V_{x}V_{x}\suf U_{x}$. By compactness of $K$ there exists $x_{1},...,x_{n}\in K$ such that $K\suf \cups{n }{i=1}x_{j}V_{x_j}$ so pick $j$ depending on $x$ such that $x_{j}\inv x\in V_{x_j}$. Let $y\in V=\caps{n }{i=1}V_{x_j}$ then $xy=x_{j}(x_{j}\inv x)y\in x_{j}U_{x_j}$ and
Since $V$ is symmetric a similar argument works for $xy\in K$.$\square$
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