Weighted metric for local uniform convergence ($\sigma (f,g)={\sum }_n{w}_n\varphi (\Vert f-g{\Vert }_{K_n})$)

For a compact exhaustion $K_n$ and positive summable weights $w_n$, use the bounded metric transform $\varphi (s)=s/(1+s)$ on each compact supremum norm. The resulting series defines a metric on continuous functions and realizes local uniform convergence on compact subsets. The triangle inequality is termwise. For convergence, control finitely many compact seminorms and then the uniformly bounded weight tail; conversely a vanishing metric controls each fixed seminorm.