Locally uniform convergence

A sequence $f_n:X\to Y$ converges locally uniformly to $f$ when every point has a neighbourhood on which the convergence is uniform.

The compact-open topology on a space of functions is generated by uniform control on compact subsets. For scalar-valued functions, its basic seminorms are $f\mapsto {\sup }_{x\in K}|f(x)|$ as $K$ ranges over compact subsets of the domain.

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.

A locally uniform limit of continuous real-valued functions is continuous. Moreover, local uniform convergence is uniform on every compact subset: choose finitely many neighbourhoods from the local uniformity cover and take the largest of their convergence indices.