Definition:Locally Uniform Convergence/General Definition

Definition
Let $X$ be a topological space.

Let $M$ be a metric space.

Let $(f_n)$ be a sequence of mappings $f_n : X\to M$.

Then $f_n$ converges locally uniformly to $f:X\to M$ if every point of $X$ has a neighborhood on which $f_n$ converges uniformly to $f$.

Also see

 * Definition:Compact Convergence