= Weighted metric for local uniform convergence
{title2=$\sigma(f,g)=\sum_n w_n\phi(\|f-g\|_{K_n})$}
For a compact exhaustion $K_n$ and positive summable weights $w_n$, use the <bounded metric transform> $\phi(s)=s/(1+s)$ 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.
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