Proposition 5.3.7.label Let $X$ be a compact metrizable space, $Y$ a locally convex space over $K\in \curl{\R,\com}$ then $C(X;Y)=UC(X;Y)$.
Proof. Let $f\in C(X;Y)$ and $\rho$ a continuous seminorm on $Y$. Consider
\begin{align*}g:X\ti X\to K&&(x,y)\mto \rho(f(x)-f(y))\end{align*}
then $g$ is continuous as a composition of continuous functions. Since $X$ is compact, so is $X\ti X$ thus $g$ is uniformly continuous as a map between metrizable spaces. Since $\rho$ was arbitrary we conclude $f$ is uniformly continuous.$\square$